A BIJECTION BETWEEN TYPE D_{n}^{(1)} CRYSTALS AND RIGGED CONFIGURATIONS

ANNE SCHILLING

Abstract. Hatayama et al. conjectured fermionic formulas associated with tensor products of U′_{q}(𝔤)-crystals B^{r,s}. The crystals B^{r,s} correspond to the Kirillov–Reshetikhin modules which are certain finite dimensional U′_{q}(𝔤)-modules. In this paper we present a combinatorial description of the affine crystals B^{r,1} of type D_{n}^{(1)}. A statistic preserving bijection between crystal paths for these crystals and rigged configurations is given, thereby proving the fermionic formula in this case. This bijection reflects two different methods to solve lattice models in statistical mechanics: the corner-transfer-matrix method and the Bethe Ansatz.

1. Introduction

The quantized universal enveloping algebra U_{q}(𝔤) associated with a symmetrizable Kac–Moody Lie algebra 𝔤 was introduced independently by Drinfeld [3] and Jimbo [6] in their study of two dimensional solvable lattice models in statistical mechanics. The parameter q corresponds to the temperature of the underlying model. Kashiwara [8] showed that at zero temperature or q = 0 the representations of U_{q}(𝔤) have bases, which he coined crystal bases, with a beautiful combinatorial structure and favorable properties such as uniqueness and stability under tensor products.

The irreducible finite-dimensional U′_{q}(𝔤)-modules were classified by Chari and Pressley [1, 2] in terms of Drinfeld polynomials. The Kirillov–Reshetikhin modules W^{r,s}, labeled by a Dynkin node r and a positive integer s, form a special class of these finite-dimensional modules. They naturally correspond to the weight sΛ_{r}, where Λ_{r} is the r-th fundamental weight of 𝔤. Recently, Hatayama et al. [5, 4] conjectured that the Kirillov–Reshetikhin modules W^{r,s} have a crystal basis denoted by B^{r,s}. The existence of such crystals allows the definition of one dimensional configuration sums, which play an important role in the study of phase transitions of two dimensional exactly solvable lattice models. For 𝔤 of type A_{n}^{(1)}, the existence of the crystal B^{r,s} was settled in [11], and the one dimensional configuration sums contain the Kostka polynomials, which arise in the theory of symmetric functions, combinatorics, the study of subgroups of finite abelian groups, and Kazhdan– Lusztig theory. In certain limits they are branching functions of integrable highest weight modules.

In [5, 4] fermionic formulas for the one dimensional configuration sums were conjectured. Fermionic formulas originate in the Bethe Ansatz of the underlying exactly solvable lattice model. The term fermionic formula was coined by the Stony Brook group [14, 15], who interpreted fermionic-type formulas for characters and branching functions of conformal field theory models as partition functions of quasiparticle systems with “fractional” statistics obeying Pauli’s exclusion principle. For type A_{n}^{(1)} the fermionic formulas were proven in [17] using a generalization of a bijection between crystals and rigged configurations of Kirillov and Reshetikhin [16]. In [19] similar bijections were used to prove the fermionic formula for nonexceptional types for crystals B^{1,1}. Rigged configurations are combinatorial objects which label the solutions to the Bethe equations. The bijection between crystals and rigged configurations reflects two different methods to solve lattice models in statistical mechanics: the corner-transfer-matrix method and the Bethe Ansatz.

Date: June 2004.

Partially supported by the NSF grant DMS-0200774.

The theory of virtual crystals [18, 20] provides a realization of crystals of type X as crystals of type Y, based on well-known natural embeddings X ↪ Y of affine algebras [7]:

\begin{matrix}C_{n}^{(1)}, A_{2n}^{(2)}, A_{2n}^{(2)†}, D_{n+1}^{(2)} & ↪ & A_{2n−1}^{(1)} \\ A_{2n−1}^{(2)}, B_{n}^{(1)} & ↪ & D_{n+1}^{(1)} \\ E_{6}^{(2)}, F_{4}^{(1)} & ↪ & E_{6}^{(1)} \\ D_{4}^{(3)}, G_{2}^{(1)} & ↪ & D_{4}^{(1)}. \end{matrix}

Note that under these embeddings every affine Kac–Moody algebra is embedded into one of simply-laced type A_{n}^{(1)}, D_{n}^{(1)} or E_{6}^{(1)}. Hence, by the virtual crystal method the combinatorial structure of any finite-dimensional affine crystal can be expressed in terms of the combinatorial crystal structure of the simply-laced types. Whereas the affine crystals B^{r,s} of type A_{n}^{(1)} are already well-understood [21], this is not the case for B^{r,s} of types D_{n}^{(1)} and E_{6}^{(1)}.

In this paper we discuss the affine crystals B^{r,1} for type D_{n}^{(1)} and the corresponding rigged configurations. The crystal structure of B^{r,1} of type D_{n}^{(1)} is given by Koga [13]. However, here we need a different combinatorial description of these crystals compatible with rigged configurations. We present a bijection between crystal paths and rigged configurations of type D_{n}^{(1)}, thereby proving the fermionic formulas of [5, 4] for type D_{n}^{(1)} corresponding to tensor products of crystals of the form B^{r,1}.

The paper is structured as follows. In Section 2 the main definitions regarding crystals are reviewed. Section 3 deals with the finite-dimensional crystals of type D_{n}^{(1)} explicitly. In particular B^{k,1} as a set is given by single column tableaux of height k with the action of the Kashiwara operators as specified in Theorem 3.2. The bijection between crystal paths and rigged configurations is stated in Section 4. The proof of the main Theorem 4.1 is relegated to Appendix A. In Section 5 statistics are defined on both crystal paths and rigged configurations. It is shown in Theorem 5.1 that the bijection preserves the statistics. This yields a proof of the fermionic formulas as presented in Section 5.1. Appendices B and C are devoted to proofs of various lemmas. In Appendix D the combinatorial R-matrix is given explicitly for highest weight elements.

Acknowledgments. I am grateful to Mark Shimozono for sharing with me the formulation of ∗-dual crystals for general types presented in Section 3.6 (see also [23]) and to Lipika Deka for her collaboration in the early stages of this project [22]. I would also like to thank the Max-Planck-Institut für Mathematik in Bonn where part of this work was carried out.

2. Crystal bases

2.1. Axiomatic definition of crystals. Let 𝔤 be a symmetrizable Kac-Moody algebra, P the weight lattice, I the index set for the vertices of the Dynkin diagram of 𝔤, {α_{i} ∈ P | i ∈ I} the simple roots, and {h_{i} ∈ P^{∗} | i ∈ I} the simple coroots. Let U_{q}(𝔤) be the quantized universal enveloping algebra of 𝔤 [10]. A U_{q}(𝔤)-crystal is a nonempty set B equipped with maps wt : B → P and e_{i}, f_{i} : B → B ∪ {∅} for all i ∈ I, satisfying

(2.1)  f_{i}(b) = b′ ⇔ e_{i}(b′) = b if b, b′ ∈ B
(2.2)  wt(f_{i}(b)) = wt(b) − α_{i} if f_{i}(b) ∈ B
(2.3)  ⟨h_{i} , wt(b)⟩ = φ_{i}(b) − ϵ_{i}(b).

Here for b ∈ B

\begin{aligned}ϵ_{i}(b) & = max\{n ≥ 0 | e_{i}^{n}(b) ≠ ∅\} \\ φ_{i}(b) & = max\{n ≥ 0 | f_{i}^{n}(b) ≠ ∅\}. \end{aligned}

(It is assumed that φ_{i}(b), ϵ_{i}(b) < ∞ for all i ∈ I and b ∈ B.) A U_{q}(𝔤)-crystal B can be viewed as a directed edge-colored graph (the crystal graph) whose vertices are the elements of B, with a directed edge from b to b′ labeled i ∈ I, if and only if f_{i}(b) = b′.

2.2. Tensor products of crystals. Let B_{1}, B_{2}, …, B_{L} be U_{q}(𝔤)-crystals. The Cartesian product B_{L} × ⋯ × B_{2} × B_{1} has the structure of a U_{q}(𝔤)-crystal using the so-called signature rule. The resulting crystal is denoted B = B_{L} ⊗ ⋯ ⊗ B_{2} ⊗ B_{1} and its elements (b_{L}, …, b_{1}) are written b_{L} ⊗ ⋯ ⊗ b_{1} where b_{j} ∈ B_{j}. The reader is warned that our convention is opposite to that of Kashiwara [10]. Fix i ∈ I and b = b_{L} ⊗ ⋯ ⊗ b_{1} ∈ B. The i-signature of b is the word consisting of the symbols + and − given by

\underbrace{− ⋯ −}_{φ_{i}(b_{L}) times} \underbrace{+ ⋯ +}_{ϵ_{i}(b_{L}) times} ⋯ \underbrace{− ⋯ −}_{φ_{i}(b_{1}) times} \underbrace{+ ⋯ +}_{ϵ_{i}(b_{1}) times}.

The reduced i-signature of b is the subword of the i-signature of b, given by the repeated removal of adjacent symbols +− (in that order); it has the form

\underbrace{− ⋯ −}_{φ times} \underbrace{+ ⋯ +}_{ϵ times}.

If φ = 0 then f_{i}(b) = ∅; otherwise

f_{i}(b_{L} ⊗ ⋯ ⊗ b_{1}) = b_{L} ⊗ ⋯ ⊗ b_{j+1} ⊗ f_{i}(b_{j}) ⊗ ⋯ ⊗ b_{1}

where the rightmost symbol − in the reduced i-signature of b comes from b_{j}. Similarly, if ϵ = 0 then e_{i}(b) = ∅; otherwise

e_{i}(b_{L} ⊗ ⋯ ⊗ b_{1}) = b_{L} ⊗ ⋯ ⊗ b_{j+1} ⊗ e_{i}(b_{j}) ⊗ ⋯ ⊗ b_{1}

where the leftmost symbol + in the reduced i-signature of b comes from b_{j}. It is not hard to verify that this well-defines the structure of a U_{q}(𝔤)-crystal with φ_{i}(b) = φ and ϵ_{i}(b) = ϵ in the above notation, with weight function

(2.4)  wt(b_{L} ⊗ ⋯ ⊗ b_{1}) = ∑_{j=1}^{L} wt(b_{j}).

This tensor construction is easily seen to be associative. The case of two tensor factors is given explicitly by

(2.5)  f_{i}(b_{2} ⊗ b_{1}) = \begin{cases}f_{i}(b_{2}) ⊗ b_{1} & if ϵ_{i}(b_{2}) ≥ φ_{i}(b_{1}) \\ b_{2} ⊗ f_{i}(b_{1}) & if ϵ_{i}(b_{2}) < φ_{i}(b_{1}) \end{cases}

and

(2.6)  e_{i}(b_{2} ⊗ b_{1}) = \begin{cases}e_{i}(b_{2}) ⊗ b_{1} & if ϵ_{i}(b_{2}) > φ_{i}(b_{1}) \\ b_{2} ⊗ e_{i}(b_{1}) & if ϵ_{i}(b_{2}) ≤ φ_{i}(b_{1}). \end{cases}

⋯  ^{\overline{n}}  ^{\overline{n-1}}  ⋯  ^{\overline{2}}  ^{\overline{1}}

Figure 1. Crystal B(ω_{1}) of the vector representation

2.3. Combinatorial R-matrix. Let B_{1} and B_{2} be two U_{q}(𝔤)-crystals such that B_{1} ⊗ B_{2} and B_{2} ⊗ B_{1} are connected and such that there is a weight λ ∈ P with unique elements u ∈ B_{1} ⊗ B_{2} and ũ ∈ B_{2} ⊗ B_{1} of weight wt(u) = wt(ũ) = λ. Then there is a unique crystal isomorphism

R : B_{1} ⊗ B_{2} → B_{2} ⊗ B_{1},

called the combinatorial R-matrix. Explicitly, R(u) = ũ and the action of R on any other element in B_{1} ⊗ B_{2} is determined by using

(2.7)  R(e_{i}(b)) = e_{i}(R(b))  and R(f_{i}(b)) = f_{i}(R(b)).
3. The crystal B^{k,1} of type D_{n}^{(1)}

From now on we restrict our attention to crystals of type D_{n} and D_{n}^{(1)}.

3.1. Dynkin data. For type D_{n}, the simple roots are

\begin{aligned}α_{i} & = ε_{i} − ε_{i+1} for 1 ≤ i < n \\ α_{n} & = ε_{n−1} + ε_{n} \end{aligned}

and the fundamental weights are

Λ_{i} = ε_{1} + ⋯ + ε_{i}  for 1 ≤ i ≤ n − 2
Λ_{n−1} = (ε_{1} + ⋯ + ε_{n−1} − ε_{n})/2
Λ_{n} = (ε_{1} + ⋯ + ε_{n−1} + ε_{n})/2

where ε_{i} ∈ Z^{n} is the i-th unit standard vector. We also define

ω_{i} = Λ_{i} = ε_{1} + ⋯ + ε_{i}  for 1 ≤ i ≤ n − 2
ω_{n−1} = Λ_{n−1} + Λ_{n} = ε_{1} + ⋯ + ε_{n−1}
ω_{n} = 2Λ_{n} = ε_{1} + ⋯ + ε_{n}
\overline{ω}_{n} = 2Λ_{n−1} = ε_{1} + ⋯ + ε_{n−1} − ε_{n}.

3.2. The crystals of antisymmetric tensor representations. The U_{q}(D_{n})-crystal graphs of the antisymmetric tensor representations [12] are denoted by B(ω_{ℓ}) (resp. B(\overline{ω}_{n})). The crystal graph B(ω_{1}) of the vector representation is given in Figure 1. The crystal B(ω_{ℓ}) (resp. B(\overline{ω}_{n})) is the connected component of B(ω_{1})^{⊗ℓ} (resp. B(ω_{1})^{⊗n}) containing the element ℓ ⊗ ℓ − 1 ⊗ ⋯ ⊗ 1 (resp. \overline{n} ⊗ n − 1 ⊗ ⋯ ⊗ 1).

Explicitly, the elements of B(ω_{ℓ}) for 1 ≤ ℓ < n are single columns of height ℓ with entries from the alphabet \{1 < 2 < ⋯ < n − 1 < n ; \overline{n} < \overline{n − 1} < ⋯ < \overline{1}\}. In Young tableau notation a column is represented by

m_{1} ; m_{2} ; ⋮ ; m_{ℓ}  which we abbreviate by m_{ℓ} … m_{1}.

If m_{ℓ} … m_{1} is in B(ω_{ℓ}) it has to satisfy

1. (m_{j}, m_{j+1}) = (n, \overline{n}), (\overline{n}, n) or m_{j} < m_{j+1} for all j;
2. If m_{a} = p and m_{b} = \overline{p}, then dist(\overline{p}, p) ≤ p.

(3.1)

Here dist(\overline{p}, p) = ℓ + 1 + a − b if m_{a} = p and m_{b} = \overline{p}.

The elements of B(ω_{n}) (resp. B(\overline{ω}_{n})) are single columns of height n. An element m_{n} … m_{1} ∈ B(ω_{n}) (resp. B(\overline{ω}_{n})) satisfies (3.1) and in addition

3. If m_{j} = n, then n − j is even (resp. odd); if m_{j} = \overline{n}, then n − j is odd (resp. even).

(3.2)

The action of e_{i}, f_{i} for 1 ≤ i ≤ n is given by embedding B(ω_{ℓ}) ↪ B(ω_{1})^{⊗ℓ} (resp. B(\overline{ω}_{n}) ↪ B(ω_{1})^{⊗n}) in the natural way by mapping m_{ℓ} ⋯ m_{1} to m_{ℓ} ⊗ ⋯ ⊗ m_{1} and using the tensor product rules.

3.3. The crystals of spinor representations. There are two crystals B(Λ_{n}) and B(Λ_{n−1}) associated with the spinor representations of D_{n} [12]. The elements of B(Λ_{n}) (resp. B(Λ_{n−1})) are given by single columns m_{n} … m_{1} such that

1. m_{j} < m_{j+1};

2. m and \overline{m} cannot occur simultaneously;

3. If m_{j} = n, then n − j is even (resp. odd); if m_{j} = \overline{n}, then n − j is odd (resp. even).

(3.3)

If b ∈ B(Λ_{n}) or B(Λ_{n−1}) and b contains i and \overline{i + 1} for 1 ≤ i < n, then f_{i}b is obtained by replacing i by i + 1 and \overline{i + 1} by \overline{i}. Else f_{i}b = 0. If b ∈ B(Λ_{n}) or B(Λ_{n−1}) and b contains n − 1 and n, then f_{n}b is obtained by replacing n − 1 by \overline{n} and n by \overline{n − 1}. Otherwise f_{n}b = 0.

Note that for k = n − 1, n there is the following embedding of crystals

B(Λ_{k}) ↪ B(2Λ_{k})

where e_{i} and f_{i} act as e_{i}^{2} and f_{i}^{2}, respectively.

3.4. Affine crystals. As a classical crystal the affine crystal B^{k,1} is isomorphic to

(3.4) B^{k,1} ≅ \begin{cases}B(Λ_{k}) ⊕ B(Λ_{k−2}) ⊕ ⋯ ⊕ B(0) & if k is even, 1 ≤ k ≤ n − 2 \\ B(Λ_{k}) ⊕ B(Λ_{k−2}) ⊕ ⋯ ⊕ B(Λ_{1}) & if k is odd, 1 ≤ k ≤ n − 2 \\ B(Λ_{k}) & if k = n − 1, n. \end{cases}

The rule for f_{0} on B^{k,1} for k = n, n − 1 is as follows. If b = \overline{1}\overline{2}m_{n−2} … m_{1} then f_{0}b = m_{n−2} … m_{1}21. Otherwise f_{0}b = 0. The operator f_{0} on B^{k,1} when 1 ≤ k < n − 1 was given by Koga [13] in terms of a tensor product of two spinor representations. For our purposes, we will need a combinatorial description of f_{0} on elements b ∈ B^{k,1} represented by a column of height k. This description of b will also be used in the description of the bijection from crystals to rigged configurations.

For 1 ≤ k ≤ n−2, we want to represent B^{k,1} as the set of all column tableaux of height k satisfying (3.1) point 1. By (3.4) B^{k,1} is the union of the sets corresponding to B(Λ_{ℓ}) with k − ℓ ≡ 0 (mod 2). We will define maps from B(Λ_{ℓ}) to column tableaux of height k. As explained in Section 3.2, b ∈ B(Λ_{ℓ}) can be represented by a column of height ℓ. If b ∈ B(Λ_{ℓ}) ⊂ B^{k,1} then fill the column of height ℓ of b successively by a pair (\overline{i}_{j}, i_{j}) for 1 ≤ j ≤ (k − ℓ)/2 in the following way to obtain a column of height k. Set i_{0} = 0. Let i_{j−1} < i_{j} be minimal such that

(1) neither i_{j} nor \overline{i_{j}} is in the column;
(2) adding i_{j} and \overline{i_{j}} to the column we have dist(\overline{i_{j}}, i_{j}) ≥ i_{j} + j;

(3) adding i_{j} and \overline{i_{j}} to the column, all other pairs (\overline{a}, a) in the new column with a > i_{j}

satisfy dist(\overline{a}, a) ≤ a + j.

It is not hard to see that f_{i} and the filling map commute, that is, the classical crystal graph with edges f_{1}, …, f_{n} does not change. Denote the filling map to height k by F_{k} or simply F.

Let D_{k} or D, the dropping map, be the inverse of F_{k}. Explicitly, given a one-column tableau b of height k, let i_{0} = 0 and successively find i_{j} > i_{j−1} minimal such that the pair (\overline{i_{j}}, i_{j}) is in b and dist(\overline{i_{j}}, i_{j}) ≥ i_{j} + j. Drop all such pairs (\overline{i_{j}}, i_{j}) from b.

For the definition of f_{0} we need slight variants of F_{k} and D_{k}, denoted by F̃_{k} and D̃_{k}, respectively, which act on columns that do not contain 1, 2, \overline{2}, \overline{1}. On these columns F̃_{k} and D̃_{k} are defined by replacing i ↦ i−2 and \overline{i} ↦ \overline{i − 2}, then applying F_{k} and D_{k}, and finally replacing i ↦ i + 2 and \overline{i} ↦ \overline{i + 2}.

Example 3.1. Let b = \overline{3}\overline{5}532 ∈ B(Λ_{5}) ⊂ B^{9,1}. Then F_{9}(b) = \overline{3567}76532. Similarly if b = \overline{2345}65421 ∈ B^{9,1} then D_{9}(b) = \overline{3}61. Finally, F̃_{5}(\overline{4}64) = \overline{45}654.

Theorem 3.2. Let b ∈ B^{k,1}. Then

(3.5)  f_{0}b = \begin{cases}F_{k}(D̃_{k−2}(x)) & if b = \overline{1}\overline{2}x \\ F̃_{k−1}(x)2 & if b = \overline{1}\overline{2}x2 \\ F̃_{k−1}(x)1 & if b = \overline{1}\overline{2}x1 \\ F̃_{k−2}(x)21 & if b = \overline{1}\overline{2}x21 \\ F_{k}(D̃_{k−1}(x)2) & if b = \overline{1}x \\ F_{k}(D̃_{k−1}(x)1) & if b = \overline{2}x \\ x21 & if b = \overline{1}x1 and D̃_{k−2}(x) = x \\ 0 & otherwise \end{cases}

(3.6)  e_{0}b = \begin{cases}F_{k}(D̃_{k−2}(x)) & if b = x21 \\ \overline{2}F̃_{k−1}(x) & if b = \overline{2}x21 \\ \overline{1}F̃_{k−1}(x) & if b = \overline{1}x21 \\ \overline{1}\overline{2}F̃_{k−2}(x) & if b = \overline{1}\overline{2}x21 \\ F_{k}(\overline{2}D̃_{k−1}(x)) & if b = x1 \\ F_{k}(\overline{1}D̃_{k−1}(x)) & if b = x2 \\ \overline{1}\overline{2}x & if b = \overline{1}x1 and D̃_{k−2}(x) = x \\ 0 & otherwise \end{cases}

where x does not contain 1, 2, \overline{2}, \overline{1}.

Proof. In [13], Koga gives the crystal B^{k,1} in terms of a tensor product of two spinor crystals. Let us denote the map from our description in terms of columns of height k to Koga’s picture by ψ. Since F_{k} commutes with all e_{i}, f_{i} for 1 ≤ i ≤ n, B^{k,1} ≅ ψ(B^{k,1}) as classical crystals. Hence ψ is specified by its action on highest weight vectors which is given by

ψ(\overline{ℓ + 1} ⋯ \overline{p} p ⋯ ℓ + 1 ℓ ⋯ 21) = \overline{ℓ + 1} ⋯ \overline{n} ℓ ⋯ 21 ⊗ n ⋯ 21

where p = (k + ℓ)/2.

Note that f_{0} commutes with f_{i} for 3 ≤ i ≤ n. Hence, to show that the above action of f_{0} on B^{k,1} is equivalent to the action given by Koga, it suffices to show that ψ(f_{0}b) = f_{0}ψ(b) where b is highest weight with respect to e_{i} for 3 ≤ i ≤ n, that is e_{i}b = 0 for these i. Call such b J′-highest weight where J′ = {3, 4, …, n}. This can be done explicitly for each case. □

Alternatively, the action of f_{0} and e_{0} is given by f_{0} = σ ∘ f_{1} ∘ σ and e_{0} = σ ∘ e_{1} ∘ σ where σ is an involution on B^{k,1} based on the type D_{n}^{(1)} Dynkin diagram automorphism that interchanges nodes 0 and 1. If b ∈ B^{k,1} contains either 1 or \overline{1}, σ(b) is obtained from b by interchanging 1 and \overline{1}. If b = \overline{1}x1, then σ(b) = F̂_{k}(D(x)). If b contains neither 1 nor \overline{1} then σ(b) = F_{k}(D̂(b)). Here D̂ and F̂ act on columns not containing 1 and \overline{1} and are defined in the same way as D̃ and F̃ with the maps i ↦ i ± 2, \overline{i} ↦ \overline{i ± 2} replaced by i ↦ i ± 1, \overline{i} ↦ \overline{i ± 1}.

For later use, it will be convenient to define the sets B̂^{k,1} for 1 ≤ k ≤ n and ˆ B̄^{n,1}. For 1 ≤ k ≤ n − 2, B̂^{k,1} = B^{k,1}. B̂^{n−1,1} is the set of all single columns of height n − 1 satisfying condition (3.1) point 1. B̂^{n,1} (resp. ˆ B̄^{n,1}) is the set of all single columns of height n satisfying (3.1) point 1 and (3.2). An affine crystal structure can be defined on B̂^{n−1,1}, B̂^{n,1} and ˆ B̄^{n,1} by extending the rules of Theorem 3.2.

Theorem 3.3. As affine crystals

\begin{aligned}B̂^{n−1,1} & ≅ B^{n,1} ⊗ B^{n−1,1} \\ B̂^{n,1} & ≅ B^{n,1} ⊗ B^{n,1} \\ ˆB̄^{n,1} & ≅ B^{n−1,1} ⊗ B^{n−1,1}. \end{aligned}

Proof. The proof is similar to the proof of Theorem 3.2. □

There are affine crystal embeddings

(3.7)  emb_{B} : B^{k,1} ↪ B^{k,1} ⊗ B^{k,1}

mapping u(B^{k,1}) ↦ u(B^{k,1}) ⊗ u(B^{k,1}) for 1 ≤ k ≤ n and sending e_{i} and f_{i} to e_{i}^{2} and f_{i}^{2}, respectively, for 0 ≤ i ≤ n. Here u(B^{k,1}) = k(k − 1) ⋯ 1 for 1 ≤ k ≤ n − 2, u(B^{n−1,1}) = \overline{n}(n − 1) ⋯ 21 and u(B^{n,1}) = n(n − 1) ⋯ 21. Note that the elements in the image of this embedding are aligned (see [18, Section 6.4]), meaning in this case that φ_{i}(b) and ϵ_{i}(b) are even for all 0 ≤ i ≤ n. Denote the image of B^{k,1} under this embedding by E^{k,1} = emb_{B}(B^{k,1}). For k = n − 1, n we view E^{n−1,1} ⊂ ˆ B̄^{n,1} and E^{n,1} ⊂ B̂^{n,1}.

As classical crystals

\begin{aligned}B̂^{n−1,1} & ≅ B(ω_{n−1}) ⊕ B(ω_{n−3}) ⊕ B(ω_{n−5}) ⊕ ⋯ \\ B̂^{n,1} & ≅ B(ω_{n}) ⊕ B(ω_{n−2}) ⊕ B(ω_{n−4}) ⊕ ⋯ \\ ˆB̄^{n,1} & ≅ B(\overline{ω}_{n}) ⊕ B(ω_{n−2}) ⊕ B(ω_{n−4}) ⊕ · · · . \end{aligned}

3.5. Paths. Let B = B^{k_{L},1} ⊗ B^{k_{L−1},1} ⊗ ⋯ ⊗ B^{k_{1},1} be a type D_{n}^{(1)} crystal with 1 ≤ k_{i} ≤ n. For a dominant integral weight λ define the set of paths as follows

P(λ, B) = {b ∈ B | wt(b) = λ, e_{i}(b) = ∅ for all i ∈ J}

where J = I\{0} = {1, 2, …, n}.

3.6. Dual crystals. Let ω_{0} be the longest element in the Weyl group of D_{n}. The action of ω_{0} on the weight lattice P of D_{n} is given by

\begin{aligned}ω_{0}(Λ_{i}) & = −Λ_{σ(i)} \\ ω_{0}(α_{i}) & = −α_{σ(i)} \end{aligned}

where σ : J → J is the identity if n is even and interchanges n − 1 and n and fixes all other Dynkin nodes if n is odd. The automorphism σ extends to the Dynkin diagram of D_{n}^{(1)} by defining σ(0) = 0.

There is a unique involution ∗ : B → B, called dual map, satisfying

\begin{aligned}wt(b^{∗}) & = ω_{0}wt(b) \\ e_{i}(b)^{∗} & = f_{σ(i)}(b^{∗}) \\ f_{i}(b)^{∗} & = e_{σ(i)}(b^{∗}). \end{aligned}

Let u(B) be the highest weight element in the highest component of B (for B = B^{k,1} with 1 ≤ k ≤ n − 2 the highest component is B(Λ_{k})). Uniqueness of ∗ follows from the connectedness of B and the fact that u(B) is the unique vector of its weight. We have

(B_{1} ⊗ B_{2})^{∗} ≅ B_{2} ⊗ B_{1}

with (b_{1} ⊗ b_{2})^{∗} ↦ b_{2}^{∗} ⊗ b_{1}^{∗}.

Explicitly, on B(Λ_{1}) the involution ∗ is given by

i ⟷ \overline{i}

except for i = n with n odd in which case n ↔ n and \overline{n} ↔ \overline{n}. A column m_{k} ⋯ m_{1} ∈ B^{k,1} is mapped to m_{1}^{∗} ⋯ m_{k}^{∗}.

4. Bijection between paths and rigged configurations

Let B = B^{k_{L},1}⊗B^{k_{L−1},1}⊗⋯⊗B^{k_{1},1} be a type D_{n}^{(1)} crystal and λ a dominant integral weight. In this section we define the set of rigged configurations RC(λ, B) and a bijection Φ : RC(λ, B) → P(λ, B).

4.1. Rigged configurations. Denote by L^{a} the number of factors B^{a,1} in B, so that B = ⨂_{a=1}^{n}(B^{a,1})^{⊗L^{a}}. For later use, let us also define L̂^{n−1} (resp. L̂^{n}, ˆ L̄^{n}) to be the number of tensor factors B̂^{n−1,1} (resp. E^{n,1}, E^{n−1,1}) in B. Let

(4.1)  L = ∑_{a∈J} L^{a}Λ_{a} + ∑_{a∈\{n−1,n\}} L̂^{a}ω_{a} + ˆ L̄^{n}\overline{ω}_{n}.

Then the sequence of partitions ν^{•} = (ν^{(1)}, ν^{(2)}, …, ν^{(n)}) is a (λ, B)-configuration if

(4.2)  ∑_{\begin{matrix}a∈J \\ i≥0 \end{matrix}} im_{i}^{(a)}α_{a} = L − λ,

where m_{i}^{(a)} is the number of parts of length i in partition ν^{(a)}. A (λ, B)-configuration is admissible if p_{i}^{(a)} ≥ 0 for all a ∈ J and i ≥ 0, where p_{i}^{(a)} is the vacancy number

(4.3)  p_{i}^{(a)} = (α_{a} | L − ∑_{b∈J} α_{b} ∑_{j≥1} min(i, j)m_{j}^{(b)}),

where (· | ·) is the normalized invariant form on P such that (α_{i} | α_{j}) is the Cartan matrix. Denote the set of admissible (λ, B)-configurations by C(λ, B). A rigged configuration (ν^{•}, J^{•}) is a sequence of partitions ν^{•} = {ν^{(a)}}_{a∈J} ∈ C(λ, B) together with a double sequence of partitions J^{•} = \{J^{(a,i)}\}\begin{matrix}a∈J \\ i≥0 \end{matrix} such that the partition J^{(a,i)} is contained in a box

of size m_{i}^{(a)} × p_{i}^{(a)}. The set of rigged configurations is denoted by RC(λ, B).

4.2. The bijection Φ. Let B′ be a tensor product of type D_{n}^{(1)} crystals of the form B^{a,1} or B̂^{n−1,1}.

Theorem 4.1. There exists a unique family of bijections Φ_{B} : RC(λ, B) → P(λ, B) such that:

(1) Suppose B = B^{1,1} ⊗ B′. Then the diagram
RC(λ, B)  ⟶^{Φ_{B}}  P(λ, B)
δ↓  ↓^{lh}
⋃_{λ^{−}} RC(λ^{−}, B′) ⟶^{Φ_{B′}} ⋃_{λ^{−}} P(λ^{−}, B′)

commutes.

(2) Suppose B = B̂^{k,1} ⊗ B′ with 2 ≤ k ≤ n − 1 and let B̃ = B^{1,1} ⊗ B̂^{k−1,1} ⊗ B′. Then the diagram

RC(λ, B) ⟶^{Φ_{B}} P(λ, B)
tj↓  ↓^{ts}
RC(λ, B̃) ⟶^{ΦB̃} P(λ, B̃)

commutes.

(3) Suppose B = B^{k,1} ⊗ B′ with k = n − 1, n. Then the diagram

RC(λ, B)  ⟶^{Φ_{B}}  P(λ, B)
δ_{s}↓  ↓^{lh}_{s}
⋃_{λ^{−s}} RC(λ^{−_{s}}, B′) ⟶^{Φ_{B′}} ⋃_{λ^{−s}} P(λ^{−_{s}}, B′)

commutes.

A proof of this theorem is given in Appendix A. In the following we define the various yet undefined maps.

4.3. Various maps. The map lh : P(λ, B^{1,1} ⊗ B′) → ⋃_{λ^{−}} P(λ^{−}, B′) in point 1 of Theorem 4.1 is given by b ⊗ b′ ↦ b′. The union is over all λ^{−} such that λ − λ^{−} is the weight of an element in B^{1,1}.

Similarly, the map lh_{s} : P(λ, B) → ⋃_{λ^{−s}} P(λ^{−_{s}}, B′) of point 3 is given by b⊗b′ ↦ b′. The union is over all λ^{−_{s}} such that λ−λ^{−_{s}} is the weight of an element in B^{k,1} for k = n−1 or n.

The map top split ts : P(λ, B) → P(λ, B̃) in point 2 of the theorem is obtained by sending m_{k}m_{k−1} ⋯ m_{1} ⊗b′ ∈ P(λ, B̂^{k,1} ⊗B′) to m_{k} ⊗m_{k−1} ⋯ m_{1} ⊗b′ ∈ P(λ, B^{1,1} ⊗ B̂^{k−1,1}⊗B′) for 1 ≤ k ≤ n−1. Similarly we may also define ts to map m_{n}m_{n−1} ⋯ m_{1}⊗ b′ ∈ P(λ, E^{n−1,1} ⊗ B′) or P(λ, E^{n,1} ⊗ B′) to m_{n} ⊗ m_{n−1} ⋯ m_{1} ⊗ b′ ∈ P(λ, B^{1,1} ⊗ B̂^{n−1,1} ⊗ B′).

Define tj : RC(λ, B) → RC(λ, B̃) in the following way. Let (ν^{•}, J^{•}) ∈ RC(λ, B). If B = B̂^{k,1}⊗B′ for 1 ≤ k ≤ n−1, tj(ν^{•}, J^{•}) is obtained from (ν^{•}, J^{•}) by adding a singular string of length 1 to each of the first k − 1 rigged partitions of ν^{•}. For B = E^{n,1} ⊗ B′, add a singular string of length 1 to (ν^{•}, J^{•})^{(a)} for 1 ≤ a ≤ n − 1. For B = E^{n−1,1} ⊗ B′, add a singular string of length 1 to (ν^{•}, J^{•})^{(n)} and (ν^{•}, J^{•})^{(a)} for 1 ≤ a ≤ n − 2.

Lemma 4.2. tj is a well-defined injection that preserves the vacancy numbers.

Proof. To show that tj is well-defined, we need to check that (ν̃^{•}, J̃^{•}) = tj(ν^{•}, J^{•}) ∈ RC(λ, B̃) for (ν^{•}, J^{•}) ∈ RC(λ, B). Let us first consider the case B = B̂^{k,1} ⊗ B′. In this case m̃_{i}^{(a)} = m_{i}^{(a)} + χ(1 ≤ a < k)δ_{i,1} and L̃ = L + ω_{1} + ω_{k−1} − ω_{k}. Hence the change of both the left and right hand side of (4.2)is ∑_{1≤a≤k−1} α_{a} = ω_{1} + ω_{k−1} − ω_{k} = ϵ_{1} − ϵ_{k}. Therefore (4.2) holds for (ν̃^{•}, J̃^{•}).

Using (4.3) the vacancy numbers transform as

p̃_{i}^{(a)} = p_{i}^{(a)} + (α_{a} | ω_{1} + ω_{k−1} − ω_{k} − ∑_{1≤b<k} α_{b}) = p_{i}^{(a)},

which shows that they are preserved under tj.

The cases B = E^{n,1} ⊗ B′ and B = E^{n−1,1} ⊗ B′ are analogous. □

The maps δ and δ_{s} are defined in the next two subsections.

4.4. Algorithm for δ. Let (ν^{•}, J^{•}) ∈ RC(λ, B^{1,1} ⊗ B′). The map δ : RC(λ, B^{1,1} ⊗ B′) → ⋃_{λ^{−}} RC(λ^{−}, B′) is defined by the following algorithm [19]. The partition J^{(a,i)} is called singular if it has a part of size p_{i}^{(a)}.

Set ℓ^{(0)} = 1 and repeat the following process for a = 1, 2, …, n − 2 or until stopped. Find the smallest index i ≥ ℓ^{(a−1)} such that J^{(a,i)} is singular. If no such i exists, set b = a and stop. Otherwise set ℓ^{(a)} = i and continue with a + 1.

If the process has not stopped at a = n − 2, find the minimal indices i, j ≥ ℓ^{(n−2)} such that J^{(n−1,i)} and J^{(n,j)} are singular. If neither i nor j exist, set b = n − 1 and stop. If i exists, but not j, set ℓ^{(n−1)} = i, b = n and stop. If j exists, but not i, set ℓ^{(n)} = j, b = \overline{n} and stop. If both i and j exist, set ℓ^{(n−1)} = i, ℓ^{(n)} = j and continue with a = n − 2.

Now continue for a = n − 2, n − 3, …, 1 or until stopped. Find the minimal index i ≥ \overline{ℓ}^{(a+1)} where \overline{ℓ}^{(n−1)} = max(ℓ^{(n−1)}, ℓ^{(n)}) such that J^{(a,i)} is singular (if i = ℓ^{(a)} then there need to be two parts of size p_{i}^{(a)} in J^{(a,i)}). If no such i exists, set b = \overline{a + 1} and stop. If the process did not stop, set b = \overline{1}. Set all yet undefined ℓ^{(a)} and \overline{ℓ}^{(a)} to ∞.

The rigged configuration (ν̃^{•}, J̃^{•}) = δ(ν^{•}, J^{•}) is obtained by removing a box from the selected strings and making the new strings singular again. Explicitly

m_{i}^{(a)}(ν̃^{•}) = m_{i}^{(a)}(ν^{•}) + \begin{cases}1 & if i = ℓ^{(a)} − 1 \\ −1 & if i = ℓ^{(a)} \\ 1 & if i = \overline{ℓ}^{(a)} − 1 and 1 ≤ a ≤ n − 2 \\ −1 & if i = \overline{ℓ}^{(a)} and 1 ≤ a ≤ n − 2 \\ 0 & otherwise. \end{cases}

The partition J̃^{(a,i)} is obtained from J^{(a,i)} by removing a part of size p_{i}^{(a)}(ν^{•}) for i = ℓ^{(a)} and i = \overline{ℓ}^{(a)}, adding a part of size p_{i}^{(a)}(ν̃^{•}) for i = ℓ^{(a)} − 1 and i = \overline{ℓ}^{(a)} − 1, and leaving it unchanged otherwise.

Example 4.3. Let (ν^{•}, J^{•}) ∈ RC(λ, B) with λ = Λ_{2} and B = (B^{1,1})^{⊗2} ⊗ (B^{2,1})^{⊗3} of type D_{4}^{(1)} given by

0  0 0  0 0  0  0 0 1  1 1  0  0 0  0  _{0  0}

Here the vacancy number p_{i}^{(a)} is written on the left of the parts of length i in ν^{(a)} and the partition J^{(a,i)} is given by the labels on the right of the parts of length i in ν^{(a)}. In this case ℓ^{(1)} = ℓ^{(2)} = ℓ^{(3)} = 1, ℓ^{(4)} = 3, \overline{ℓ}^{(2)} = \overline{ℓ}^{(1)} = ∞ and b = \overline{3}, so that δ(ν^{•}, J^{•}) is

0  0 0  0  0  0 0 1  1  _{1  0}  _{1  1} .

Altogether

Φ_{B}(ν^{•}, J^{•}) = \overline{3} ⊗ \overline{4} ⊗ 4 3 ⊗ \overline{1} 1 ⊗ 2 1 .

4.5. Algorithm for δ_{s}. The embedding (3.7) can be extended to paths

(4.4)  emb_{P} : P(λ, B) ↪ P(λ̂, B̂)

where λ̂ = 2λ, B = ⨂_{a=1}^{n}(B^{a,1})^{⊗L^{a}}, and B̂ = ⨂_{a=1}^{n}(B^{a,1})^{⊗L̂^{a}} where L̂^{a} = 2L^{a} for 1 ≤ a ≤ n. A path b = b_{L} ⊗ ⋯ ⊗ b_{1} ∈ P(λ, B) is mapped to emb_{P}(b) = emb_{B}(b_{L}) ⊗ ⋯ ⊗ emb_{B}(b_{1}).

An analogous embedding can be defined on rigged configurations

(4.5)  emb_{RC} : RC(λ, B) ↪ RC(λ̂, B̂)

where emb_{RC}(ν^{•}, J^{•}) = 2(ν^{•}, J^{•}), meaning that all parts in ν^{(a)} and J^{(a,i)} are doubled in length.

Theorem 4.4. For B a tensor product of crystals of the form B^{a,1} with 1 ≤ a ≤ n − 2 the following diagram commutes:

RC(λ, B) ⟶^{Φ_{B}} P(λ, B)
emb_{RC}↓  ↓^{emb}_{B}
RC(λ̂, B̂) ⟶^{ΦB̂} P(λ̂, B̂).

The proof of this theorem is given in Appendix B.

The definition of δ_{s} is given in such a way that Theorem 4.4 also holds for the spinor case. Let us define

δ_{s} : RC(λ, B) → ⋃_{λ^{−s}} RC(λ^{−_{s}}, B′)

as follows. For (ν^{•}, J^{•}) ∈ RC(λ, B) apply emb_{RC} and then a sequence of points 1 and 2 of Theorem 4.1 to remove the last tensor factor. The claim is that this rigged configuration is in the image of emb_{RC} so that one can apply emb_{RC}^{−1}. The result is δ_{s}(ν^{•}, J^{•}). The proof of the well-definedness of δ_{s} is part of the proof of Theorem 4.1 as given in Appendix A.

Example 4.5. Let (ν^{•}, J^{•}) ∈ RC(λ, B) with λ = 2Λ_{1} + Λ_{4} and B = B^{5,1} ⊗ B^{2,1} ⊗ (B^{1,1})^{⊗3} of type D_{5}^{(1)} given by

2  1  0  0 0  0  0  0 0  0  _{0  0}  _{0  0}

Then δ_{s}(ν^{•}, J^{•}) ∈ RC(λ^{−_{s}}, B′) with λ^{−_{s}} = Λ_{1} + Λ_{2} and B′ = B^{2,1} ⊗ (B^{1,1})^{⊗3} is

1  1  0  0 0  0  0 0  _{0  0}  _{0  0}

The details are given in Table 1. The first entry is emb_{RC}(ν^{•}, J^{•}). The next entries are obtained by acting with δ ∘ tj. Acting with emb_{RC}^{−1} on the last rigged configuration yields δ_{s}(ν^{•}, J^{•}). In the last column of the table we recorded b of the algorithm δ. Hence the step in B^{5,1} corresponding to this rigged configuration is \overline{2}\overline{5}431.

5. Statistics on paths and rigged configurations

A statistic can be defined on both paths and rigged configurations. In this section we define the intrinsic energy function on paths and cocharge statistics on rigged configurations and show that the bijection Φ preserves the statistics (see Theorem 5.1). This gives rise to the fermionic formula presented in Section 5.1.

In Section 2.3 we defined the combinatorial R-matrix R : B_{2} ⊗ B_{1} → B_{1} ⊗ B_{2}. In addition, there exists a function H : B_{2} ⊗ B_{1} → ℤ called the local energy function, that is unique up to a global additive constant. It is constant on J components and satisfies for all b_{2} ∈ B_{2} and b_{1} ∈ B_{1} with R(b_{2} ⊗ b_{1}) = b′_{1} ⊗ b′_{2}

H(e_{0}(b_{2} ⊗ b_{1})) = H(b_{2} ⊗ b_{1}) + \begin{cases}−1 & if ϵ_{0}(b_{2}) > φ_{0}(b_{1}) and ϵ_{0}(b′_{1}) > φ_{0}(b′_{2}) \\ 1 & if ϵ_{0}(b_{2}) ≤ φ_{0}(b_{1}) and ϵ_{0}(b′_{1}) ≤ φ_{0}(b′_{2}) \\ 0 & otherwise. \end{cases}

We shall normalize the local energy function by the condition H(u(B_{2}) ⊗ u(B_{1})) = 0.

(ν̃^{•}, J̃^{•})^{(1)}  (ν̃^{•}, J̃^{•})^{(2)}  (ν̃^{•}, J̃^{•})^{(3)} (ν̃^{•}, J̃^{•})^{(4)} (ν̃^{•}, J̃^{•})^{(5)} b
4  2 0  0 0  0 0  0 0  0 _{0  0} _{0  0}
3  2 0  0 0  0  0  0 0  0 _{1  0} _{0  0} \overline{2}
3  2 0  0 0  0  0  0 1  0 _{0  0}  _{0  0} \overline{5}
3  2 0  0 1  0  0  0 0  _{0  0}  _{0  0} 4
3  2 0  0 0  0  0 0  _{0  0}  _{0  0} 3
2  2 0  0 0  0  0 0  _{0  0}  _{0  0} 1

Table 1. An example for δ_{s}

For a crystal B^{k,1} of type D_{n}^{(1)}, the intrinsic energy D_{B^{k,1}} : B^{k,1} → ℤ is defined as follows. Let b ∈ B^{k,1} which is in the classical component B(Λ_{k−2j}) (see (3.4)). Then D_{B^{k,1}}(b) = j.

On the tensor product B = B_{L} ⊗ ⋯ ⊗ B_{1} of simple crystals there is an intrinsic energy function defined D_{B} : B → ℤ (see for example [20, Section 2.5])

(5.1)  D_{B} = ∑_{1≤i<j≤L} H_{i}R_{i+1}R_{i+2} ⋯ R_{j−1} + ∑_{j=1}^{L} D_{B_{j}}π_{1}R_{1}R_{2} ⋯ R_{j−1}.

Here R_{i} and H_{i} denote the combinatorial R-matrix and local energy function acting on the i-th and (i + 1)-th tensor factors, respectively. π_{1} is the projection onto the rightmost tensor factor.

Similarly, there is a statistic on the set of rigged configurations given by cc(ν^{•}, J^{•}) = cc(ν^{•}) + ∑_{a,i} |J^{(a,i)}| where |J^{(a,i)}| is the size of the partition J^{(a,i)} and

(5.2)  cc(ν^{•}) = \frac{1}{2} ∑_{a,b∈J} ∑_{j,k≥1}(α_{a} | α_{b}) min(j, k)m_{j}^{(a)}m_{k}^{(b)}.

Let Φ̃ = Φ∘θ where θ : RC(λ, B) → RC(λ, B) with θ(ν^{•}, J^{•}) = (ν^{•}, J̃^{•}) is the function which complements the riggings, meaning that J̃^{•} is obtained from J^{•} by complementing all partitions J^{(a,i)} in the m_{i}^{(a)} × p_{i}^{(a)} rectangle.

Theorem 5.1. Let B = B^{k_{L},1} ⊗ ⋯ ⊗ B^{k_{1},1} be a crystal of type D_{n}^{(1)} and λ a dominant integral weight. The bijection Φ̃ : RC(λ, B) → P(λ, B) preserves the statistics, that is cc(ν^{•}, J^{•}) = D(Φ̃(ν^{•}, J^{•})) for all (ν^{•}, J^{•}) ∈ RC(λ, B).

A proof of this theorem is given in Section 5.4 using the results of Sections 5.2 and 5.3.

Example 5.2. For the rigged configuration of Example 4.3 we have

Φ̃(ν^{•}, J^{•}) = \overline{3} ⊗ \overline{4} ⊗ \overline{1} 1 ⊗ 4 3 ⊗ 2 1

and cc(ν^{•}, J^{•}) = D(Φ̃(ν^{•}, J^{•})) = 10.

5.1. Fermionic formulas. An immediate corollary of Theorems 4.1 and 5.1 is the equality

∑_{b∈P(λ,B)} q^{D(b)} = ∑_{(ν^{•},J^{•})∈RC(λ,B)} q^{cc(ν^{•},J^{•})}.

The left-hand side is in fact the one dimensional configuration sum

X(λ, B) = ∑_{b∈P(λ,B)} q^{D(b)}.

The right-hand side can be simplified slightly by observing that the generating function of partitions in a box of width p and height m is the q-binomial coefficient

\left[{m + p \atop m}\right] = \frac{(q)_{p+m}}{(q)_{m}(q)_{p}} ,

where (q)_{m} = (1 − q)(1 − q^{2}) ⋯ (1 − q^{m}). Hence the right-hand side becomes the fermionic formula

M(λ, B) = ∑_{ν^{•}∈C(λ,B)} q^{cc(ν^{•})} ∏_{\begin{matrix}i≥1 \\ a∈J \end{matrix}} \left[{m_{i}^{(a)} + p_{i}^{(a)} \atop m_{i}^{(a)}}\right]

where m_{i}^{(a)} and p_{i}^{(a)} are as defined in Section 4.1. This proves the following result conjectured in [5, 4].

Theorem 5.3. For 𝔤 of type D_{n}^{(1)}, λ a dominant weight and B = B^{k_{L},1} ⊗ ⋯ ⊗ B^{k_{1},1}

X(λ, B) = M(λ, B).

5.2. Relations between the various maps. The maps ts, tj, lh, δ were defined in Sections 4.3 and 4.4. We may define similar maps using the dualities ∗ and θ.

For crystals let rh = ∗ ∘ lh ∘ ∗ and bs = ∗ ∘ ts ∘ ∗. Explicitly, rh(b′ ⊗ b) = b′ for b′ ⊗b ∈ B′ ⊗B^{1,1} and bs(b′ ⊗m_{k} ⋯ m_{2}m_{1}) = b′ ⊗m_{k} ⋯ m_{2} ⊗m_{1} for b′ ⊗m_{k} ⋯ m_{1} ∈ B′ ⊗ B̂^{k,1} for 2 ≤ k ≤ n − 1. Similarly bs(b′ ⊗ m_{n} ⋯ m_{2}m_{1}) = b′ ⊗ m_{n} ⋯ m_{2} ⊗ m_{1} for b′ ⊗ m_{n} ⋯ m_{1} ∈ B′ ⊗ E^{n−1,1} or E^{n,1}.

On the rigged configuration side define δ̃ = θ ∘ δ ∘ θ and bj = θ ∘ tj ∘ θ. Explicitly, bj adds parts of lengths one to the same rigged partitions as tj; however in this case they are not singular, but of label 0.

In the following [·, ·] denotes the commutator.

Lemma 5.4. We have [bj, δ] = 0 and [tj, δ̃] = 0.

Proof. The relation [tj, δ̃] = 0 follows from [bj, δ] = 0 by conjugation by θ. Hence it suffices to prove [bj, δ] = 0.

Let (ν^{•}, J^{•}) ∈ RC(λ, B) where B = B̂^{1,1} ⊗ B̂^{k,1} ⊗ B′. We assume throughout the proof that 1 ≤ k < n. The cases B = B̂^{1,1} ⊗ E^{n,1} ⊗ B′ and B = B̂^{1,1} ⊗ E^{n−1,1} ⊗ B′ can be proven in a similar fashion. Let ℓ^{(a)} and \overline{ℓ}^{(a)} (resp. s^{(a)} and \overline{s}^{(a)}) be the singular strings selected by δ in (ν^{•}, J^{•}) (resp. bj(ν^{•}, J^{•})). Clearly, s^{(a)} ≤ ℓ^{(a)} and \overline{s}^{(a)} ≤ \overline{ℓ}^{(a)} since bj changes neither vacancy numbers nor riggings by Lemma 4.2 and its definition bj = θ ∘ tj ∘ θ. If s^{(a)} = ℓ^{(a)} and \overline{s}^{(a)} = \overline{ℓ}^{(a)} for all a, the lemma holds.

Assume that b is minimal such that s^{(b)} < ℓ^{(b)}. Certainly we must have 1 ≤ b < k, s^{(a)} = ℓ^{(a)} = 1 for 1 ≤ a < b, s^{(b)} = 1 < ℓ^{(b)}, m_{1}^{(b)} = 0 and p_{1}^{(b)} = 0. One may easily verify that (see also [17, Eq. (3.10)])

\begin{matrix}−p_{i−1}^{(a)} + 2p_{i}^{(a)} − p_{i+1}^{(a)} = & − ∑_{b∈J} (α_{a} | α_{b}) m_{i}^{(b)} + δ_{i,1}L^{a} \\ ≥ & − ∑_{b∈J} (α_{a} | α_{b}) m_{i}^{(b)}, \end{matrix}

where δ_{i,j} = 1 if i = j and 0 else. Using m_{1}^{(b)} = 0 and p_{1}^{(b)} = 0 this reads for i = 1 and a = b

−p_{2}^{(b)} = m_{1}^{(b−1)} + m_{1}^{(b+1)} + L^{b}.

Since p_{i}^{(a)} ≥ 0 for all admissible rigged configurations, this implies that all quantities in this equation must be zero. For b > 1 the condition m_{1}^{(b−1)} = 0 contradicts ℓ^{(b−1)} = 1. For b = 1 we have L^{1} ≥ 1 which makes p_{2}^{(1)} ≥ 0 impossible. Hence s^{(a)} = ℓ^{(a)} for all a as desired. The proof that \overline{s}^{(a)} = \overline{ℓ}^{(a)} is analogous. □

Lemma 5.5. We have [δ, δ̃] = 0.

The proof of Lemma 5.5 is outlined in Appendix C.

We say that f corresponds to g under Φ if the following diagram commutes:

RC(λ, B) ⟶^{Φ_{B}} P(λ, B)
f↓  ↓^{g}
RC(λ′, B′) ⟶_{Φ_{B′}} P(λ′, B′).

Proposition 5.6. The following maps correspond under Φ:

(1) bs and bj

(2) ts and tj

(3) lh and δ

(4) rh and δ̃

(5) ∗ and θ

(6) R and id

Proof. Points (2) and (3) hold by definition.

We prove (1) by induction on B. For simplicity we drop the dependency on λ in all diagrams to follow. Also for brevity we denote B̂^{k,1} ⊗ B ⊗ B̂^{k′,1} symbolically by B . The splitting B̂^{k,1} → B̂^{k−1,1} ⊗ B̂^{1,1} will be denoted by → □ □ and so on. Consider the following diagram:

(5.3) \xymatrix{RC( B ) \ar[rrr]^{Φ} \ar[ddd]_{bj} \ar[dr]^{tj} &  &  & P( B ) \ar[ddd]^{bs} \ar[dl]_{ts} \\  & RC(□ □B ) \ar[r]^{Φ} \ar[d]_{bj} & P(□ □B ) \ar[d]^{bs} \\  & RC(□ □B□ □) \ar[r]_{Φ} & P(□ □B□ □) \\ RC( B□ □) \ar@{-->}[ur]_{tj} \ar[rrr]_{Φ} &  &  & P( B□ □) \ar@{-->}[ul]^{ts}}

This diagram can be viewed as a cube with the front face given by the large square. The back face commutes by induction. It is easy to check that the left and right faces commute. The top and bottom face commute by (2). Hence it follows from [17, Lemma 5.3] that the front face commutes since ts is injective.

Next consider the diagram:

(5.4)  \xymatrix{RC(□B ) \ar[rrr]^{Φ} \ar[ddd]_{bj} \ar[dr]^{δ} &  &  & P(□B ) \ar[ddd]^{bs} \ar[dl]_{lh} \\  & RC(B ) \ar[r]^{Φ} \ar[d]_{bj} & P(B ) \ar[d]^{bs} \\  & RC(B□ □) \ar[r]_{Φ} & P(B□ □) \\ RC(□B□ □) \ar[ur]_{δ} \ar[rrr]_{Φ} &  &  & P(□B□ □) \ar[ul]^{lh}}

The back face commutes by induction, the top and bottom faces commute by the definition of Φ. It is easy to check that the right face commutes. The left face commutes by Lemma 5.4. Hence the front face commutes by [17, Lemma 5.3]. Diagrams (5.3)and (5.4)together prove (1).

For the proof of (4) we use the same abreviated notation as before. B̂^{1,1} is symbolically denoted by □. Consider the following diagram:

(5.5)  \xymatrix{RC(□B□) \ar[rrr]^{Φ} \ar[ddd]_{δ̃} \ar[dr]^{δ} &  &  & P(□B□) \ar[ddd]^{rh} \ar[dl]_{lh} \\  & RC(B□) \ar[r]^{Φ} \ar[d]_{δ̃} & P(B□) \ar[d]^{rh} \\  & RC(B) \ar[r]_{Φ} & P(B) \\ RC(□B) \ar@{-->}[ur]_{δ} \ar[rrr]_{Φ} &  &  & P(□B) \ar[ul]^{lh}}

The back face commutes by induction and the top and bottom face commute by (3). It is easy to verify that the right face commutes. The left face commutes by Lemma 5.5. Hence the front face commutes, since lh is injective for fixed initial weight λ.

Next consider the diagram:

(5.6)  [figure: RC( B□); Φ; δ̃; tj; P( B□); rh; ts; RC(□ □B□); Φ; δ̃; P(□ □B□); rh; RC(□ □B); Φ; P(□ □B); RC( B); tj; Φ; P( B); ts]

The back face commutes by induction, the top and bottom face commute by (2). It is easy to verify that the right face commutes. The left face commutes by Lemma 5.4. Hence the front face commutes, since ts is injective. Diagrams (5.5) and (5.6) together prove that (4) holds.

For the proof of point (5) consider the following diagram:

(5.7)  \xymatrix{RC(□B) \ar[rrr]^{Φ} \ar[ddd]_{θ} \ar[dr]^{δ} &  &  & P(□B) \ar[ddd]^{∗} \ar[dl]_{lh} \\  & RC(B) \ar[r]^{Φ} \ar[d]_{θ} & P(B) \ar[d]^{∗} \\  & RC(B^{∗}) \ar[r]_{Φ} & P(B^{∗}) \\ RC(B^{∗}□) \ar[ur]_{δ̃} \ar[rrr]_{Φ} &  &  & P(B^{∗}□) \ar[ul]^{rh}}

The back face commutes by induction. The left and right faces commute by definition. The top and bottom faces commute by (3) and (4), respectively. Since δ̃ is injective, the front face commutes by [17, Lemma 5.3].

Next consider the following diagram:

(5.8)  \xymatrix{RC( B) \ar[rrr]^{Φ} \ar[ddd]_{θ} \ar@{-->}[dr]^{tj} &  &  & P( B) \ar[ddd]^{∗} \ar[dl]_{ts} \\  & RC(□ □B) \ar[r]^{Φ} \ar[d]_{θ} & P(□ □B) \ar[d]^{∗} \\  & RC(B^{∗}□ □) \ar[r]_{Φ} & P(B^{∗}□ □) \\ RC(B^{∗} ) \ar@{-->}[ur]_{bj} \ar[rrr]_{Φ} &  &  & P(B^{∗} ) \ar@{-->}[ul]^{bs}}

The back face commutes by induction. The left and right faces commute by definition. The bottom and top faces commute by (1) and (2), respectively. Since bj is injective, the front face commutes by [17, Lemma 5.3]. Hence (5) holds by diagrams (5.7) and (5.8).

For the proof of (6) we may assume that R = R_{i} is the combinatorial R-matrix acting on the i-th and (i + 1)-th tensor factor. By induction we may assume that i = L − 1 where L is the number of tensor factors in B. By part (5) we may assume that i = 1. Again by induction we may assume that L = 2, which means that B = B^{k_{2},1} ⊗ B^{k_{1},1}. Hence, since R is a crystal isomorphism, it suffices to check (6) for highest weight elements in B = B^{k′,1} ⊗ B^{k,1}. The combinatorial R-matrix R : B^{k′,1} ⊗ B^{k,1} → B^{k,1} ⊗ B^{k′,1} for all highest weight elements is given explicitly in Appendix D. It is tedious but straightforward to check (6) for these. □

5.3. Properties of the statistics. In this section we show how the statistics cc and D change under tj and bs, respectively. These results will be used in the next section in the proof of Theorem 5.1. For the next two lemmas we assume that B is the tensor product of crystals B̂^{k,1} for 1 ≤ k < n, E^{n−1,1} or E^{n,1}.

Lemma 5.7. For (ν^{•}, J^{•}) ∈ RC(λ, B) we have

cc(tj(ν^{•}, J^{•})) − cc(ν^{•}, J^{•}) = \begin{cases}(ϵ_{1} − ϵ_{k} | L) + 1 & if B = B̂^{k,1} ⊗ B′ for 1 ≤ k < n \\ (ϵ_{1} − ϵ_{n} | L) + 1 & if B = E^{n,1} ⊗ B′ \\ (ϵ_{1} + ϵ_{n} | L) + 1 & if B = E^{n−1,1} ⊗ B′. \end{cases}

Proof. First consider the case B = B̂^{k,1} ⊗ B′. Then by (5.2)

cc(tj(ν^{•})) =\frac{1}{2} ∑_{a,b} ∑_{i,j}(α_{a} | α_{b}) min(i, j)
× (m_{i}^{(a)} + χ(i = 1)χ(a < k))(m_{j}^{(b)} + χ(j = 1)χ(b < k))
=cc(ν^{•}) + ∑_{b,j}(α_{1} + ⋯ + α_{k−1} | α_{b})m_{j}^{(b)} + 1.

Since tj adds a singular string of length one to the first k − 1 rigged partitions, the change from the labels is, using (4.3),

p_{1}^{(1)} + p_{1}^{(2)} + ⋯ + p_{1}^{(k−1)} = (α_{1} + ⋯ + α_{k−1} | L − ∑_{b,j} α_{b}m_{j}^{(b)}).

Hence

cc(tj(ν^{•}, J^{•})) − cc(ν^{•}, J^{•}) = (ϵ_{1} − ϵ_{k} | L) + 1.

The cases B = E^{n,1} ⊗ B′ and B = E^{n−1,1} ⊗ B′ are similar. □

Lemma 5.8. For p ∈ P(λ, B) we have

D(bs(p)) − D(p) = \begin{cases}(ϵ_{1} − ϵ_{k} | L) + 1 & if B = B′ ⊗ B̂^{k,1} for 1 ≤ k < n \\ (ϵ_{1} − ϵ_{n} | L) + 1 & if B = B′ ⊗ E^{n,1} \\ (ϵ_{1} + ϵ_{n} | L) + 1 & if B = B′ ⊗ E^{n−1,1}. \end{cases}

Proof. It follows from the structure of (5.1) that we can restrict ourselves to the case when B consists of one or two tensor factors. Let b ∈ B̂^{k,1} (E^{n−1,1} or E^{n,1}) and b′ ∈ B̂^{k′,1} (E^{n−1,1} or E^{n,1}) such that e_{i}(b) = 0 and e_{i}(b′ ⊗ b) = 0 for 1 ≤ i ≤ n. Furthermore, write bs(b) = b_{2} ⊗ b_{1} ∈ B̂^{k−1,1} ⊗ B^{1,1} (B̂^{n−1,1} ⊗ B^{1,1}) and

\begin{aligned}R(b′ ⊗ b) & = b̂ ⊗ b̂′ \\ R(b′ ⊗ b_{2}) & = • ⊗ b̃′ \\ R(b̃′ ⊗ b_{1}) & = • ⊗ ˜ b̃′, \end{aligned}

where the crystal elements denoted • are not of interest to us here. Let L′ be the weight corresponding to B̂^{k′,1} (resp. E^{n−1,1} or E^{n,1}) (see (4.1)). Then it suffices to prove the following two equations:

D(b_{2} ⊗ b_{1}) − D(b) = 1 (5.9)

H(b′ ⊗ b_{2}) + H(b̃′ ⊗ b_{1}) + D(˜b̃′) − H(b′ ⊗ b) − D(b̂′) (5.10)

= \begin{cases}(ϵ_{1} − ϵ_{k} | L′) & if b ∈ B̂^{k,1} \\ (ϵ_{1} − ϵ_{n} | L′) & if b ∈ E^{n,1} \\ (ϵ_{1} + ϵ_{n} | L′) & if b ∈ E^{n−1,1}. \end{cases}

To prove (5.9) and (5.10) we use the same notation as in Appendix D. Namely, a column with entries ⋯ a_{4} ⋯ a_{3}a_{2} ⋯ a_{1}, where a_{2} ⋯ a_{1} denotes a sequence of consecutive integers a_{2} a_{2} − 1 a_{2} − 2 … a_{1} + 1 a_{1}, is compactly written as [a_{1}..a_{2} | a_{3}..a_{4} | ⋯].

Let us first concentrate on (5.9) for b ∈ B̂^{k,1}. Since e_{i}(b) = 0 for 1 ≤ i ≤ n, b is of the form b = [1..p | \overline{p}..\overline{ℓ}]. Then b_{2}⊗b_{1} = [2..p | \overline{p}..\overline{ℓ}]⊗[1] and R(b_{2}⊗b_{1}) = b̃_{1}⊗b̃_{2} = [ℓ−1]⊗ [1..p−1 | \overline{p − 1}..\overline{ℓ − 1}]. Hence D(b) = p−ℓ+1 and D(b_{2} ⊗b_{1}) = H(b_{2} ⊗b_{1})+D(b_{1})+ D(b̃_{2}) = 1 + 0 + (p − ℓ + 1) = p − ℓ + 2 which proves (5.9). The highest weight vector in E^{n,1} is b = [1..n] and b_{2}⊗b_{1} = [2..n]⊗[1], so that R(b_{2}⊗b_{1}) = b̃_{1}⊗b̃_{2} = [n]⊗[1..n−1]. In this case D(b) = 0 and D(b_{2} ⊗ b_{1}) = H(b_{2} ⊗ b_{1}) + D(b_{1}) + D(b̃_{2}) = 1 + 0 + 0 = 1. The highest weight vector in E^{n−1,1} is b = [1..n − 1 \overline{n}] and the calculation is analogous to the case E^{n,1}.

Equation (5.10) can be proven for each case listed in Appendix D, where it should be noted that the results for B^{k′,1} ⊗ B^{k,1} with 1 ≤ k′ ≤ k ≤ n − 2 can be extended to B̂^{k′,1} ⊗ B̂^{k,1} with 1 ≤ k′ ≤ k ≤ n − 1 and B̂^{k′,1} ⊗ B̂^{n,1} (B̂^{k′,1} ⊗ ˆ B̄^{n,1}) with k′ < n.

Here we show the details for Case 1 for k′ < k ≤ n and r ≤ ℓ. In this case b′ ⊗ b = [1..ℓ | k + 1..p | \overline{p}..\overline{q} | \overline{ℓ}..\overline{r}] ⊗ [1..k], b_{2} = [2..k] and b_{1} = [1]. Also by the results of Appendix D, H(b′ ⊗ b) = k′ − ℓ and D(b̂′) = 0. To determine H(b′ ⊗ b_{2}), H(b̃′ ⊗ b_{1}) and D(˜b̃′), note that

e_{q}e_{q+1} ⋯ e_{n−2} e_{n}e_{n−1} ⋯ e_{1}(b′ ⊗ b_{2}) = [1..ℓ | k..p | \overline{p}..\overline{q + 1} | \overline{ℓ}..\overline{r}] ⊗ [1..k − 1].

Acting on this by the R-matrix one obtains

\begin{cases}[1..ℓ + 1 | k′ + 1..k − 2 | \overline{k − 2}..\overline{q + 1} | \overline{ℓ + 1}..\overline{r}] ⊗ [1..k′] & if p = k − 1, k − q even \\ [1..ℓ | k′ + 1..p | \overline{p}..\overline{q + 1} | \overline{ℓ}..\overline{r}] ⊗ [1..k′] & else \end{cases}

so that in both cases b̃′ = [2..k′ + 1] and ˜ b̃′ = [1..k′]. Since H is constant on classical components, H(b′ ⊗ b_{2}) = k′ − ℓ, H(b̃′ ⊗ b_{1}) = 1 and D(˜b̃′) = 0. Hence the result of (5.10) is 1 as claimed for k′ < k ≤ n.

In Appendix D, the R-matrix is only worked out for the case k′ ≤ k. To prove (5.10) for k ≤ k′, consider the equation under the duality operation ∗ which swaps tensor factors. Since ts = ∗ ∘ bs ∘ ∗, (5.10) becomes

\begin{aligned}H(b′_{2} ⊗ b) & + H(b′_{1} ⊗ b̃) + D(˜b̃) − H(b′ ⊗ b) − D(b̂) \\ & = \begin{cases}(ϵ_{1} − ϵ_{k′} | L) & if b′ ∈ B̂^{k′,1} \\ (ϵ_{1} − ϵ_{n} | L) & if b′ ∈ E^{n,1} \\ (ϵ_{1} + ϵ_{n} | L) & if b′ ∈ E^{n−1,1} \end{cases} \end{aligned} ; (5.11)

where ts(b′) = b′_{1} ⊗ b′_{2}, R(b′_{2} ⊗ b) = b̃ ⊗ •, R(b′_{1} ⊗ b̃) = ˜ b̃ ⊗ •, and L is the weight corresponding to B̂^{k,1} (resp. E^{n,1}, E^{n−1,1}). Again (5.11) can be verified for all cases of Appendix D. Here we demonstrate that it holds for Case 1 with k′ ≤ k < n and r ≤ ℓ, so that b′ ⊗ b = [1..ℓ | k + 1..p | \overline{p}..\overline{q} | \overline{ℓ}..\overline{r}] ⊗ [1..k], b′_{1} = [\overline{r}] and b′_{2} = [1..ℓ | k + 1..p | \overline{p}..\overline{q} | \overline{ℓ}..\overline{r + 1}]. Hence H(b′ ⊗ b) = k′ − ℓ and H(b′_{2} ⊗ b) = k′ − ℓ − 1. Next let us calculate D(b̂). By Appendix D, b̂ = [1..ℓ | k′ + 1..p | \overline{p}..\overline{q} | \overline{ℓ}..\overline{r}] if p ≠ k or k − q odd. Acting with a product of e_{i} with i ≠ 0 to make b̂ into a highest weight vector, one finds that D(b̂) = d + max{0, p − q + 1 − k′ + ℓ}. However, since k′ = 2ℓ + 2p − k − q − r + 2 we have p − q + 1 − k′ + ℓ = (k − p) + (r − ℓ) − 1 < 0 because k − p ≤ 0 and r − ℓ ≤ 0. This implies that D(b̂) = d. Similarly, for p = k and k − q even one finds that D(b̂) = d + 1. By Appendix D

b̃ = \begin{cases}[1..ℓ + 1 | k′..k − 1 | \overline{k − 1}..\overline{q} | \overline{ℓ + 1}..\overline{r + 1}] & if p = k, k − q even \\ [1..ℓ | k′..p | \overline{p}..\overline{q} | \overline{ℓ}..\overline{r + 1}] & else. \end{cases}

In both cases the highest weight vector obtained from b′_{1} ⊗b̃ by the application of a product of e_{i}’s with i ≠ 0 is of the form [\overline{y}] ⊗ [1..x | \overline{x}..\overline{y + 1}] where x = k − d when p = k and k − q even and x = k − d + 1 otherwise. The variable y is fixed by the requirement that k = 2x − y. By Case 5a of Appendix D we have

R([\overline{y}] ⊗ [1..x | \overline{x}..\overline{y + 1}]) = [1..x − 1 | \overline{x − 1}..\overline{y} | \overline{1}] ⊗ [1]

so that D(˜b̃) = x − y + 1 = k + 1 − x which is d + 1 if p = k and k − q even and d otherwise. Hence altogether the left-hand side of (5.11) yields 0 as expected.

Finally let us show that (5.11) yields 2 for the tensor product E^{n−1,1} ⊗ E^{n,1} and 0 for E^{n,1} ⊗ E^{n,1}. The highest weight elements in E^{n−1,1} ⊗ E^{n,1} are of the form b′ ⊗ b = [1..n − 2ℓ − 1 | \overline{n}..\overline{n − 2ℓ}] ⊗ [1..n]. By Appendix D, H(b′ ⊗ b) = 2ℓ and D(b̂) = 0. By the definition of ts, b′_{1} = [\overline{n − 2ℓ}] and b′_{2} = [1..n − 2ℓ − 1 | \overline{n}..\overline{n − 2ℓ + 1}]. Furthermore

b̃ ⊗ • = R(b′_{2} ⊗ b) = [1..n − 2ℓ − 1 | n\overline{n}..\overline{n − 2ℓ + 1}] ⊗ [1..n − 1]

with H(b′_{2} ⊗ b) = 2ℓ. The highest weight vector corresponding to b′_{1} ⊗ b̃ is [\overline{n}] ⊗ [1..n] with R([\overline{n}] ⊗ [1..n]) = [1..n − 1 | \overline{1}] ⊗ [1]. Hence H(b′_{1} ⊗ b̃) = 1 and D(˜b̃) = 1, so that (5.11) yields indeed 2.

The highest weight elements in E^{n,1} ⊗ E^{n,1} are of the form b′ ⊗ b = [1..n − 2ℓ | \overline{n}..\overline{n − 2ℓ + 1}] ⊗ [1..n]. By Appendix D, H(b′ ⊗ b) = 2ℓ and D(b̂) = 0. By the definition of ts, b′_{1} = [\overline{n − 2ℓ + 1}] and b′_{2} = [1..n − 2ℓ | \overline{n}..\overline{n − 2ℓ + 2}]. Furthermore

b̃ ⊗ • = R(b′_{2} ⊗ b) = [1..n − 2ℓ + 1 | \overline{n − 1}..\overline{n − 2ℓ + 1}] ⊗ [1..n − 1]

with H(b′_{2}⊗b) = 2ℓ−1. The highest weight vector corresponding to b′_{1}⊗b̃ is [1]⊗[1..n−1 | \overline{n − 1}] with R([1] ⊗ [1..n − 1 | \overline{n − 1}]) = [1..n − 1 | \overline{n − 1}] ⊗ [1]. Hence H(b′_{1} ⊗ b̃) = 0 and D(˜b̃) = 1, so that (5.11) yields indeed 0.

All other cases can be shown in a similar fashion. □

5.4. Proof of Theorem 5.1. By Proposition 5.6 the following diagram commutes

RC(λ, B) ⟶^{Φ} P(λ, B) ⟶^{∗} P(λ, B)
tj↓  ↓^{ts}  ↓^{bs}
RC(λ, B̃) ⟶_{Φ} P(λ, B̃) ⟶_{∗} P(λ, B̃).

Since Φ̃ = ∗ ∘ Φ, this implies that tj corresponds to bs under Φ̃.

Let (ν^{•}, J^{•}) ∈ RC(λ, B) and b = Φ̃(ν^{•}, J^{•}) ∈ P(λ, B). Using (4.4) and (4.5) define (ν̂^{•}, Ĵ^{•}) = emb_{RC}(ν^{•}, J^{•}) ∈ RC(λ̂, B̂) and b̂ = emb_{P}(b) ∈ P(λ̂, B̂) with λ̂ and B̂ as defined in Section 4.5. It is easy to see that cc(ν̂^{•}, Ĵ^{•}) = 2cc(ν^{•}, J^{•}) since all parts of the partitions and all riggings double. Similarly, by [18, Eq. (6.23)] it follows that DB̂(b̂) = 2D_{B}(b). Since the embedding map commutes with the bijection Φ̃ by Theorem 4.4 and the definition of the spinor case, we may restrict ourselves to the image of emb.

There exists a sequence S_{P} of maps bs and R which transforms b̂ into a path of single boxes, that is S_{P} : P(λ̂, B̂) → P(λ̂, (B^{1,1})^{N}) where N = ∑_{a=1}^{n} aL̂^{a} + nˆL̄^{n}. By Proposition 5.6 there is a corresponding sequence S_{RC} : RC(λ̂, B̂) → RC(λ̂, (B^{1,1})^{N}) under Φ̃ consisting of maps tj and id.

By Lemmas 5.7 and 5.8 and the fact that R and id do not change the statistics, it follows that for b̂ = Φ̃(ν̂^{•}, Ĵ^{•})

D(S_{P}(b̂)) = cc(S_{RC}(ν̂^{•}, Ĵ^{•})) implies that D(b̂) = cc(ν̂^{•}, Ĵ^{•}),

which in turn implies that D(b) = cc(ν^{•}, J^{•}). But Theorem 5.1 for the case B = (B^{1,1})^{N} has already been proven in [19].

Appendix A. Proof of Theorem 4.1

The proof proceeds by induction on B. Given a tensor product B = B^{k_{L},1} ⊗B^{k_{L−1},1} ⊗ ⋯ ⊗ B^{k_{1},1}, there is a unique sequence of transformations − : B = B^{1,1} ⊗ B′ → B′, ts : B = B^{k,1} ⊗B′ → B^{1,1} ⊗B^{k−1,1} ⊗B′ for 2 ≤ k ≤ n−2 and −_{s} : B = B^{k,1} ⊗B′ → B′ for k = n − 1, n resulting in the empty crystal. If B is empty then both RC(λ, B) and P(λ, B) are the empty set unless λ = ∅, in which case P(λ, B) is the singleton consisting of the empty path, RC(λ, B) is the singleton consisting of the empty rigged configuration, and Φ_{∅} is the unique bijection RC(∅, ∅) → P(∅, ∅).

Proof of (1). Any map Φ_{B} satisfying (1) is injective by definition and unique by induction. It was proven in [19] that − and δ are bijections.

Proof of (2). To prove the existence and uniqueness of a bijection satisfying (2), it must be shown that ΦB̃ restricts to a bijection ΦB̃ : tj(RC(λ, B)) → ts(P(λ, B)). The proof relies on two lemmas for which we need the following notation. Let (ν^{•}, J^{•}) ∈ tj(RC(λ, B)) and (ν̃^{•}, J̃^{•}) = tj ∘ δ(ν^{•}, J^{•}). Let ℓ^{(a)}, \overline{ℓ}^{(a)} be the length of the selected singular strings

in (ν^{•}, J^{•}) by δ, and ℓ̃^{(a)},˜ \overline{ℓ}^{(a)} the ones selected in (ν̃^{•}, J̃^{•}) by δ.

Lemma A.1. The following inequalities hold:

(1) ℓ̃^{(a)} ≥ ℓ^{(a+1)} for 1 ≤ a ≤ n − 3

(2) ℓ̃^{(n−2)} ≥ min(ℓ^{(n−1)}, ℓ^{(n)})

(3) ℓ̃^{(n−1)} ≥ \overline{ℓ}^{(n−2)} if ℓ^{(n−1)} = min(ℓ^{(n−1)}, ℓ^{(n)})

ℓ̃^{(n)} ≥ \overline{ℓ}^{(n−2)} if ℓ^{(n)} = min(ℓ^{(n−1)}, ℓ^{(n)})

(4) ˜ \overline{ℓ}^{(a)} ≥ \overline{ℓ}^{(a−1)} for 2 ≤ a ≤ n − 1.

Similarly, let b ∈ P(λ, B), b̃ = − ∘ ts(b) and ˜ b̃ = − ∘ ts(b̃). More precisely, if b = m_{k}m_{k−1} ⋯ m_{1} ⊗ b′, then b̃ = m_{k−1}m_{k−2} ⋯ m_{1} ⊗ b′ and ˜ b̃ = m_{k−2} ⋯ m_{1} ⊗ b′. By induction (ν̃^{•}, J̃^{•}) = Φ^{−1}(b̃) and (˜ν̃^{•}, ˜ J̃^{•}) = Φ^{−1}(˜b̃) exist. Let s̃^{(a)}, ˜ \overline{s}^{(a)} be the lengths of the selected strings by δ^{−1} in (˜ν̃^{•}, ˜ J̃^{•}) adding m_{k−1} ∈ B^{1,1} to the path, and s^{(a)}, \overline{s}^{(a)} the length of the selected strings in (ν̃^{•}, J̃^{•}) adding the step m_{k} ∈ B^{1,1} to the path.

Lemma A.2. The following inequalities hold:

(1) \overline{s}^{(a)} ≤ ˜ \overline{s}^{(a+1)} for 1 ≤ a ≤ n − 3
(2) \overline{s}^{(n−2)} ≤ max(s̃^{(n−1)}, s̃^{(n)})
(3) s^{(n−1)} ≤ s̃^{(n−2)} if s̃^{(n−1)} = max(s̃^{(n−1)}, s̃^{(n)})
s^{(n)} ≤ s̃^{(n−2)} if s̃^{(n)} = max(s̃^{(n−1)}, s̃^{(n)})
(4) s^{(a+1)} ≤ s̃^{(a)} for 0 ≤ a ≤ n − 3.

Proof of Lemma A.1. Writing p_{i}^{(a)}(ν^{•}) to indicate the dependence of the vacancy numbers on the configuration, one obtains from the algorithm for δ and (4.3) that for 1 ≤ a ≤ n − 3

\begin{aligned}p_{i}^{(a)}(ν̃^{•}) = p_{i}^{(a)}(ν^{•}) & − χ(ℓ^{(a−1)} ≤ i < ℓ^{(a)}) + χ(ℓ^{(a)} ≤ i < ℓ^{(a+1)}) \\ & + χ(\overline{ℓ}^{(a)} ≤ i < \overline{ℓ}^{(a−1)}) − χ(\overline{ℓ}^{(a+1)} ≤ i < \overline{ℓ}^{(a)}) \\ p_{i}^{(n−2)}(ν̃^{•}) = p_{i}^{(n−2)}(ν^{•}) & − χ(ℓ^{(n−3)} ≤ i < ℓ^{(n−2)}) + χ(ℓ^{(n−2)} ≤ i < min(ℓ^{(n−1)}, ℓ^{(n)})) \\ & + χ(\overline{ℓ}^{(n−2)} ≤ i < \overline{ℓ}^{(n−3)}) − χ(max(ℓ^{(n−1)}, ℓ^{(n)}) ≤ i < \overline{ℓ}^{(n−2)}) \\ p_{i}^{(n−1)}(ν̃^{•}) = p_{i}^{(n−1)}(ν^{•}) & − χ(ℓ^{(n−2)} ≤ i < ℓ^{(n−1)}) + χ(ℓ^{(n−1)} ≤ i < \overline{ℓ}^{(n−2)}) \\ p_{i}^{(n)}(ν̃^{•}) = p_{i}^{(n)}(ν^{•}) & − χ(ℓ^{(n−2)} ≤ i < ℓ^{(n)}) + χ(ℓ^{(n)} ≤ i < \overline{ℓ}^{(n−2)}). \end{aligned} (A.1)

Since (ν^{•}, J^{•}) ∈ tj(RC(λ, B)), there is a singular string of length one in the first k − 1 rigged partitions, so that ℓ^{(a)} = 1 for 0 ≤ a ≤ k − 1 (recall that ℓ^{(0)} = 1 by definition).

Similarly, ℓ̃^{(a)} = 1 for 0 ≤ a ≤ k − 2. Hence ℓ̃^{(a)} = ℓ^{(a+1)} = 1 for 0 ≤ a ≤ k − 2 so that (1) holds in this case.

We proceed by induction on a. Fix k − 1 ≤ a ≤ n − 3. By induction hypothesis ℓ̃^{(a−1)} ≥ ℓ^{(a)}. By definition ℓ̃^{(a)} ≥ ℓ̃^{(a−1)}, so that ℓ̃^{(a)} ≥ ℓ^{(a)}. By (A.1), (ν̃^{•}, J̃^{•})^{(a)} has no singular string of length i in the range ℓ^{(a)} ≤ i < ℓ^{(a+1)} which implies that ℓ̃^{(a)} ≥ ℓ^{(a+1)} for 1 ≤ a ≤ n − 3. Hence (1) holds.

Next consider a = n−2. By construction, ℓ̃^{(n−2)} ≥ ℓ̃^{(n−3)} and by (1), ℓ̃^{(n−3)} ≥ ℓ^{(n−2)}. Hence ℓ̃^{(n−2)} ≥ ℓ^{(n−2)}. By (A.1), (ν̃^{•}, J̃^{•})^{(n−2)} has no singular string of length i in the range ℓ^{(n−2)} ≤ i < min(ℓ^{(n−1)}, ℓ^{(n)}). Hence ℓ̃^{(n−2)} ≥ min(ℓ^{(n−1)}, ℓ^{(n)}) which proves (2).

Consider (ν̃^{•}, J̃^{•})^{(n−1)} and (ν̃^{•}, J̃^{•})^{(n)}. By construction, ℓ̃^{(n−1)} ≥ ℓ̃^{(n−2)} and ℓ̃^{(n)} ≥ ℓ̃^{(n−2)}. Using (2) we get ℓ̃^{(n−1)} ≥ min(ℓ^{(n−1)}, ℓ^{(n)}) and ℓ̃^{(n)} ≥ min(ℓ^{(n−1)}, ℓ^{(n)}). If min(ℓ^{(n−1)}, ℓ^{(n)}) = ℓ^{(n−1)}, then ℓ̃^{(n−1)} ≥ ℓ^{(n−1)}. Now looking at the change in vacancy number (A.1) for p_{i}^{(n−1)}, we conclude that ℓ̃^{(n−1)} ≥ \overline{ℓ}^{(n−2)}. If on the other hand min(ℓ^{(n−1)}, ℓ^{(n)}) = ℓ^{(n)}, then ℓ̃^{(n)} ≥ ℓ^{(n)}. Now, by the change in vacancy number (A.1) for p_{i}^{(n)}, ℓ̃^{(n)} ≥ \overline{ℓ}^{(n−2)}. This proves (3).

We prove (4) by downward induction on a. Recall that ˜ \overline{ℓ}^{(n−1)} = max(ℓ̃^{(n−1)}, ℓ̃^{(n)}) by

definition. Using (3) we obtain ˜ \overline{ℓ}^{(n−1)} ≥ \overline{ℓ}^{(n−2)}. This starts the induction at a = n−1. Fix

2 ≤ a ≤ n − 2 and assume that ˜ \overline{ℓ}^{(a+1)} ≥ \overline{ℓ}^{(a)} by induction hypothesis. By construction,

˜\overline{ℓ}^{(a)} ≥ ˜ \overline{ℓ}^{(a+1)}, so that ˜ \overline{ℓ}^{(a)} ≥ \overline{ℓ}^{(a)}. By the change in vacancy number (A.1), (ν̃^{•}, J̃^{•})^{(a)} has

no singular string of length i in the range \overline{ℓ}^{(a)} ≤ i < \overline{ℓ}^{(a−1)}. This implies that ˜ \overline{ℓ}^{(a)} ≥ \overline{ℓ}^{(a−1)} as desired. □

Proof of Lemma A.2. The proof is similar to the proof of Lemma A.1. □

To show that ΦB̃ ∘ tj(RC(λ, B)) ⊂ ts(P(λ, B)), let (ν^{•}, J^{•}) ∈ tj(RC(λ, B)) and m_{k} ⊗ m_{k−1}m_{k−2} ⋯ m_{1} ⊗ b′ = ΦB̃(ν^{•}, J^{•}). We need to show that either m_{k} > m_{k−1} or (m_{k}, m_{k−1}) = (n, \overline{n}), (\overline{n}, n).

If m_{k} = p with p < n, then ℓ^{(a)} = ∞ for a ≥ p and by Lemma A.1 ℓ̃^{(a)} = ∞ for a ≥ p − 1. This implies that m_{k} > m_{k−1}.

If m_{k} = n, then ℓ^{(n−1)} < ∞, ℓ^{(n)} = ∞ and \overline{ℓ}^{(a)} = ∞ for 1 ≤ a ≤ n − 2. By Lemma

A.1 this implies ℓ̃^{(n−1)} = ∞ and ˜ \overline{ℓ}^{(a)} = ∞ for 1 ≤ a ≤ n − 2. Hence m_{k−1} = \overline{n} or m_{k−1} < m_{k}. The case m_{k} = \overline{n} is analogous.

If m_{k} = \overline{n − 1}, then ℓ^{(n−1)}, ℓ^{(n)} < ∞ and \overline{ℓ}^{(a)} = ∞ for 1 ≤ a ≤ n − 2. By Lemma A.1 it follows that either ℓ̃^{(n−1)} or ℓ̃^{(n)} is ∞ so that indeed m_{k−1} < m_{k}.

Finally, if m_{k} = \overline{p} with 1 ≤ p ≤ n − 2, then \overline{ℓ}^{(a)} = ∞ for 1 ≤ a < p. By Lemma A.1

it follows that ˜ \overline{ℓ}^{(p)} = ∞ so that again m_{k} > m_{k−1}.

Next it needs to be shown that ΦB̃ : tj(RC(λ, B)) → ts(P(λ, B)) is surjective. First consider the case B = B̂^{k,1} ⊗ B′ for 2 ≤ k ≤ n − 1. Let b = m_{k} ⊗ m_{k−1}m_{k−2} ⋯ m_{1} ⊗ b′ ∈ ts(P(λ, B)). Let (ν^{•}, J^{•}) = Φ−1 ; B̃(b). We need to show that (ν^{•}, J^{•}) ∈ tj(RC(λ, B)). In other words, it needs to be shown that (ν^{•}, J^{•})^{(a)} has a singular string of length one for 1 ≤ a ≤ k − 1. By induction on k it follows that s̃^{(a)} = 0 for 0 ≤ a ≤ k − 2 (remember that s^{(0)} = 0 by definition). Hence by Lemma A.2, s^{(a)} = 0 for 0 ≤ a ≤ k − 1 which implies that there are singular strings of lengths one in (ν^{•}, J^{•})^{(a)} for 1 ≤ a ≤ k − 1.

This completes the proof of case (2).

Proof of (3). To prove the existence and uniqueness of a bijection satisfying (3) we need to show that (a) δ_{s} is well-defined and (b) Φ_{B′} restricts to a bijection δ_{s}(RC(λ, B)) → −_{s}(P(λ, B)).

Let (ν^{•}, J^{•}) ∈ RC(λ, B) and (ν_{0}^{•}, J_{0}^{•}) = emb_{RC}(ν^{•}, J^{•}). Inductively define (ν_{j}^{•}, J_{j}^{•}) =

δ ∘ tj(ν_{j−1}^{•}, J_{j−1}^{•}) for 1 ≤ j ≤ n and let ℓ_{j}^{(a)}, \overline{ℓ}_{j}^{(a)} be the lengths of the singular strings selected by δ in tj(ν_{j}^{•}, J_{j}^{•}) for 0 ≤ j < n. For (a) it needs to be shown that (ν_{n}^{•}, J_{n}^{•}) is in the image of emb_{RC}. This follows from the following lemma:

Lemma A.3. For all 0 ≤ j < n we have:

(1) For 1 ≤ a ≤ n − 2, ℓ_{j}^{(a)} is odd, and

ℓ_{j}^{(a)} = \begin{cases}1 & if a < n − j \\ \overline{ℓ}_{j−(n−a)}^{(a)} − 1 & if a ≥ n − j. \end{cases}

In addition for i = ℓ_{j}^{(a)} − 1, p_{i}^{(a)}(ν_{j+1}^{•}) is even.

(2) For k = n and j even, or k = n − 1 and j odd we have

ℓ_{j}^{(n−1)} = ℓ_{j−1}^{(n−1)} − 1 and ℓ_{j}^{(n)} is even
min(ℓ_{j}^{(n−1)}, ℓ_{j}^{(n)}) = ℓ_{j}^{(n−1)}
p_{i}^{(n−1)}(ν_{j+1}^{•}) is even for i = ℓ_{j}^{(n−1)} − 1.

Similarly, for k = n and j odd, or k = n − 1 and j even we have

ℓ_{j}^{(n)} = ℓ_{j−1}^{(n)} − 1 and ℓ_{j}^{(n−1)} is even
min(ℓ_{j}^{(n−1)}, ℓ_{j}^{(n)}) = ℓ_{j}^{(n)}
p_{i}^{(n)}(ν_{j+1}^{•}) is even for i = ℓ_{j}^{(n)} − 1.

(3) For 1 ≤ a ≤ n − 2, \overline{ℓ}_{j}^{(a)} is even.

Lemma A.3 implies that all partitions in (ν_{n}^{•}, J_{n}^{•}) have only parts of even lengths and even riggings so that (ν_{n}^{•}, J_{n}^{•}) is indeed in the image of emb_{RC}. Hence δ_{s} is well-defined.

Proof of Lemma A.3. For concreteness we assume throughout that k = n. The case k = n − 1 is entirely analogous. For j = 0, by definition tj(ν_{0}^{•}, J_{0}^{•}) has singular strings of lengths 1 in the first n−1 rigged partitions. Hence ℓ_{0}^{(a)} = 1 for 1 ≤ a ≤ n−1. This proves (1). Furthermore tj(ν_{0}^{•}, J_{0}^{•})^{(n)} contains only even strings so that ℓ_{0}^{(n)} > ℓ_{0}^{(n−1)} = 1 is even which proves (2). All strings of length greater than one in (ν_{0}^{•}, J_{0}^{•})^{(a)} for 1 ≤ a ≤ n − 2 are even so that \overline{ℓ}_{0}^{(a)} are even which verifies (3).

Suppose the lemma is true for 0 ≤ j′ < j. We will prove it for j. Set b = n−j. Clearly ℓ_{j}^{(a)} = 1 for 1 ≤ a < b by the definition of tj. We proceed by induction on b ≤ a < n − 2.

Assume that (1) holds for p < a. In particular ℓ_{j}^{(p)} = \overline{ℓ}_{j−(n−p)}^{(p)} − 1 for b ≤ p < a. We

want to show that ℓ_{j}^{(a)} = \overline{ℓ}_{j−(n−a)}^{(a)} − 1. For a = b, we have \overline{ℓ}_{j−(n−a)}^{(a)} − 1 ≥ 1 = ℓ_{j}^{(a−1)}.

Similarly for a > b note that by Lemma A.1 \overline{ℓ}_{j−(n−a)}^{(a)} − 1 ≥ \overline{ℓ}_{j−(n−a+1)}^{(a−1)} − 1 = ℓ_{j}^{(a−1)}.

We will show that for ℓ_{j}^{(a−1)} ≤ i < \overline{ℓ}_{j−(n−a)}^{(a)} − 1 there is no singular string in (ν_{j}^{•}, J_{j}^{•})^{(a)}. For this range of i the change in vacancy number is

p_{i}^{(a)}(ν_{j}^{•}) = p_{i}^{(a)}(ν_{0}^{•}) − ∑_{k=0}^{j−1} χ(ℓ_{k}^{(a−1)} ≤ i < ℓ_{k}^{(a)}) + ∑_{k=0}^{j−1} χ(ℓ_{k}^{(a)} ≤ i < ℓ_{k}^{(a+1)})

− ∑_{k=0}^{j−1} χ(\overline{ℓ}_{k}^{(a+1)} ≤ i < \overline{ℓ}_{k}^{(a)}) + ∑_{k=0}^{j−1} χ(\overline{ℓ}_{k}^{(a)} ≤ i < \overline{ℓ}_{k}^{(a−1)})

= p_{i}^{(a)}(ν_{0}^{•}) + χ(ℓ_{j−1}^{(a)} ≤ i < ℓ_{j−1}^{(a+1)}) − χ(\overline{ℓ}_{a−b}^{(a+1)} ≤ i < \overline{ℓ}_{a−b}^{(a)}).

By induction, ℓ_{j−1}^{(a+1)} = \overline{ℓ}_{a−b}^{(a+1)} − 1, so that the vacancy numbers in this range change by ±1 except for i = ℓ_{j−1}^{(a+1)}. By induction ℓ_{j−1}^{(a+1)} is odd and ν_{j}^{(a)} has no odd parts of length \overline{ℓ}_{a−b−1}^{(a)} ≤ i < \overline{ℓ}_{a−b}^{(a)} − 1. In addition the riggings in this range are even, so that there are no singular strings for ℓ_{j}^{(a−1)} ≤ i < \overline{ℓ}_{a−b}^{(a)} − 1 as claimed. Hence ℓ_{j}^{(a)} = \overline{ℓ}_{a−b}^{(a)} − 1.

To see that p_{i}^{(a)}(ν_{j+1}^{•}) is even for i = ℓ_{j}^{(a)}−1 note that \overline{ℓ}_{a−b+m}^{(a−1)} ≥ \overline{ℓ}_{a−b}^{(a)} and \overline{ℓ}_{a−b+m+1}^{(a+1)} ≥ \overline{ℓ}_{a−b}^{(a)} for m ≥ 0. Furthermore by induction ℓ_{j−m}^{(a−1)} = \overline{ℓ}_{a−b−m−1}^{(a−1)} − 1 for m ≥ 0 and ℓ_{j−m}^{(a+1)} = \overline{ℓ}_{a−b−m+1}^{(a+1)} − 1 for m ≥ 1, so that all parts of length less than ℓ_{j}^{(a)} in ν_{j+1}^{(a−1)} and ν_{j+1}^{(a+1)} are even. Hence p_{i}^{(a)}(ν_{j+1}^{•}) is even.

For a = n − 2, we will also show that there is no singular string in (ν_{j}^{•}, J_{j}^{•})^{(n−2)} of length ℓ_{j}^{(n−3)} ≤ i < \overline{ℓ}_{j−2}^{(n−2)} − 1. Similarly to before we find that in this range of i the change in vacancy number is

p_{i}^{(n−2)}(ν_{j}^{•}) = p_{i}^{(n−2)}(ν_{0}^{•}) + χ(ℓ_{j−1}^{(n−2)} ≤ i < min(ℓ_{j−1}^{(n−1)}, ℓ_{j−1}^{(n)})) − χ(max(ℓ_{j−2}^{(n−1)}, ℓ_{j−2}^{(n)}) ≤ i < \overline{ℓ}_{j−2}^{(n−2)}).

If j is even, by induction min(ℓ_{j−1}^{(n−1)}, ℓ_{j−1}^{(n)}) = ℓ_{j−1}^{(n)}, max(ℓ_{j−2}^{(n−1)}, ℓ_{j−2}^{(n)}) = ℓ_{j−2}^{(n)} and ℓ_{j−1}^{(n)} = ℓ_{j−2}^{(n)} − 1 with ℓ_{j−2}^{(n)} is even, so that the vacancy number changes by ±1 except for i = ℓ_{j−2}^{(n)} − 1. However, there is no odd string in ν_{j}^{(n−2)} for \overline{ℓ}_{j−3}^{(n−3)} ≤ i < \overline{ℓ}_{j−2}^{(n−2)} − 1 and all riggings in this range are even. Hence ℓ_{j}^{(n−2)} = \overline{ℓ}_{j−2}^{(n−2)} − 1 as desired. The case j odd is analogous.

To verify that p_{i}^{(n−2)}(ν_{j+1}^{•}) is even for i = ℓ_{j}^{(n−2)} − 1 let us again assume that j is even. Note that ℓ_{j−1}^{(n−1)} ≥ \overline{ℓ}_{j−2}^{(n−2)} by Lemma A.1 and by induction ℓ_{j−k+1}^{(n)} = ℓ_{j−k}^{(n)} − 1 for k > 0 even and ℓ_{j−k+1}^{(n−1)} = ℓ_{j−k}^{(n−1)} − 1 for k > 1 odd. Hence ν_{j+1}^{(n−1)} and ν_{j+1}^{(n)} contain only even strings of length less than ℓ_{j}^{(n−2)}. Similarly, ν_{j+1}^{(n−3)} contains only even strings of length less than ℓ_{j}^{(n−2)} which proves the assertion. The case j odd is again analogous. This concludes the proof of (1).

For the proof of (2) assume that j is even. To prove ℓ_{j}^{(n−1)} = ℓ_{j−1}^{(n−1)} − 1 we show that there are no singular strings of length ℓ_{j}^{(n−2)} ≤ i < ℓ_{j−1}^{(n−1)} − 1 in (ν_{j}^{•}, J_{j}^{•})^{(n−1)}. By induction min(ℓ_{j−2}^{(n−1)}, ℓ_{j−2}^{(n)}) = ℓ_{j−2}^{(n−1)}, so that by Lemma A.1 ℓ_{j−1}^{(n−1)} ≥ \overline{ℓ}_{j−2}^{(n−2)} = ℓ_{j}^{(n−2)} + 1. Furthermore, ℓ_{j}^{(n−2)} ≥ ℓ_{j−1}^{(n−2)}. Hence for ℓ_{j}^{(n−2)} ≤ i < ℓ_{j−1}^{(n−1)} − 1 the vacancy number changes by

\begin{matrix}p_{i}^{(n−1)}(ν_{j}^{•}) = & p_{i}^{(n−1)}(ν_{0}^{•}) − ∑_{k=0}^{j−1} χ(ℓ_{k}^{(n−2)} ≤ i < ℓ_{k}^{(n−1)}) + ∑_{k=0}^{j−1} χ(ℓ_{k}^{(n−1)} ≤ i < \overline{ℓ}_{k}^{(n−2)}) \\ = & p_{i}^{(n−1)}(ν_{0}^{•}) − χ(ℓ_{j−1}^{(n−2)} ≤ i < ℓ_{j−1}^{(n−1)}). \end{matrix}

Since p_{i}^{(n−1)}(ν_{0}^{•}) is even and all riggings in this range are even, this implies that there are no singular strings. Hence ℓ_{j}^{(n−1)} = ℓ_{j−1}^{(n−1)} − 1 as claimed.

By induction min(ℓ_{j−1}^{(n−1)}, ℓ_{j−1}^{(n)}) = ℓ_{j−1}^{(n)}, so that by Lemma A.1 ℓ_{j}^{(n)} ≥ \overline{ℓ}_{j−1}^{(n−2)}. On the

other hand ℓ_{j}^{(n−1)} = ℓ_{j−1}^{(n−1)} − 1 < \overline{ℓ}_{j−1}^{(n−2)} which proves that min(ℓ_{j}^{(n−1)}, ℓ_{j}^{(n)}) = ℓ_{j}^{(n−1)}.

Note that ℓ_{j′}^{(n)} ≤ \overline{ℓ}_{j−1}^{(n−2)} for j′ < j so that there are only even strings of length greater or

equal to \overline{ℓ}_{j−1}^{(n−2)} in ν_{j}^{(n)}. Hence ℓ_{j}^{(n)} must be even. All strings of length less than ℓ_{j}^{(n−1)} in ν_{j+1}^{(n−2)} are even by induction. Hence p_{i}^{(n−1)}(ν_{j+1}^{•}) is even for i = ℓ_{j}^{(n−1)} − 1.

The case j is odd is analogous. This concludes the proof of (2).

For (3) note that \overline{ℓ}_{j}^{(a)} ≥ \overline{ℓ}_{j−1}^{(a−1)} by Lemma A.1. There are only even strings of length

greater or equal to \overline{ℓ}_{j−1}^{(a−1)} in ν_{j}^{(a)} which proves that \overline{ℓ}_{j}^{(a)} is even. □

To prove (b) we need to show that the crystal element defined by δ_{s} is indeed an element of B^{n,1} (resp. B^{n−1,1}) and that δ_{s}^{−1} exists and is well-defined. The proof of the latter is similar to the proof of Lemma A.3 and is omitted here. To show that the step defined by δ_{s} is in B^{n,1} (resp. B^{n−1,1}) we need to show that (3.3) holds. For concreteness let us assume that the crystal is B^{n,1}. The condition m_{h} < m_{h+1} follows from (2) of Theorem 4.1 by the definition of δ_{s}.

Let j be maximal such that both ℓ_{j−1}^{(n−1)}, ℓ_{j−1}^{(n)} < ∞. If j is even, ℓ_{j}^{(n−1)} = ℓ_{j−1}^{(n−1)} − 1 < ∞ by Lemma A.3, so that ℓ_{j}^{(n)} = ∞. This implies that m_{n−j} = n. Similarly, if j is odd, ℓ_{j}^{(n)} = ℓ_{j−1}^{(n)} − 1 < ∞ and ℓ_{j}^{(n−1)} = ∞, so that m_{n−j} = \overline{n}. This proves point 3 of (3.3). It also shows that either n or \overline{n} occurs in b = m_{n} … m_{1}. To see that not both n and \overline{n} can occur, we note that if j is even then ℓ_{j+1}^{(n)} = ℓ_{j}^{(n)} − 1 = ∞ and min(ℓ_{j+1}^{(n−1)}, ℓ_{j+1}^{(n)}) = ℓ_{j+1}^{(n)} by Lemma A.3, so that ℓ_{j+1}^{(n−1)} = ∞ as well. Hence m_{n−j−1} ≤ n − 1. The case j odd is analogous.

It remains to show that precisely one of p and \overline{p} occurs in b for 1 ≤ p ≤ n − 1. Suppose m_{n−j} = \overline{p}. Then \overline{ℓ}_{j}^{(p−1)} = ∞ and \overline{ℓ}_{j}^{(p)} < ∞. By Lemma A.1 \overline{ℓ}_{j−1}^{(p−1)} ≤ \overline{ℓ}_{j}^{(p)} < ∞ and

\overline{ℓ}_{j+1}^{(p)} ≥ \overline{ℓ}_{j}^{(p−1)} = ∞. Setting a = n − p + j it follows from Lemma A.3 that

ℓ_{a}^{(p−1)} = \overline{ℓ}_{j−1}^{(p−1)} − 1 < ∞  ℓ_{a}^{(p)} = \overline{ℓ}_{j}^{(p)} − 1 < ∞
ℓ_{a+1}^{(p−1)} = \overline{ℓ}_{j}^{(p−1)} − 1 = ∞  ℓ_{a+1}^{(p)} = \overline{ℓ}_{j+1}^{(p)} − 1 = ∞.

Hence p does not occur in b.

Now suppose \overline{p} ∉ b. We show that then p ∈ b. If \overline{p} ∉ b there must be some j such that \overline{ℓ}_{j−1}^{(p)}, \overline{ℓ}_{j−1}^{(p−1)} < ∞ and \overline{ℓ}_{j}^{(p)}, \overline{ℓ}_{j}^{(p−1)} = ∞. Setting as before a = n − p + j this implies by

Lemma A.3 that ℓ_{a}^{(p−1)} = \overline{ℓ}_{j−1}^{(p−1)} − 1 < ∞ and ℓ_{a}^{(p)} = \overline{ℓ}_{j}^{(p)} − 1 = ∞. Hence m_{n−a} = p. This proves point 2 of (3.3).

This concludes the proof of Theorem 4.1.

Appendix B. Proof of Theorem 4.4

In the following let us suppress the weight λ and write RC(B) for RC(λ, B). It is straightforward to check that the diagram

(B.1)  \xymatrix{RC(B^{k,1} ⊗ B) \ar[r]^{^{emb^{RC}}} \ar[ddd]_{δ∘tj} & RC(B^{k,1} ⊗ B^{k,1} ⊗ B̂) \ar[d]^{δ∘tj} \\  & RC(B^{k−1,1} ⊗ B^{k,1} ⊗ B̂) \ar[d]^{id} \\  & RC(B^{k,1} ⊗ B^{k−1,1} ⊗ B̂) \ar[d]^{δ∘tj} \\ RC(B^{k−1,1} ⊗ B) \ar[r]^{_{emb_{RC}}} & RC(B^{k−1,1} ⊗ B^{k−1,1} ⊗ B̂)}

commutes.

We will show here that the crystal analogon to this diagram commutes:

(B.2)  \xymatrix{B^{k,1} \ar[r]^{^{emb^{B}}} \ar[ddd]_{−∘ts} & B^{k,1} ⊗ B^{k,1} \ar[d]^{−∘ts} \\  & B^{k−1,1} ⊗ B^{k,1} \ar[d]^{R} \\  & B^{k,1} ⊗ B^{k−1,1} \ar[d]^{−∘ts} \\ B^{k−1,1} \ar[r]^{_{emb_{B}}} & B^{k−1,1} ⊗ B^{k−1,1}.}

By the definition of Φ and Proposition 5.6, the commutative diagrams (B.1) and (B.2) imply Theorem 4.4.

First we verify (B.2) for classically highest weight elements. All classically highest weight elements b ∈ B^{k,1} are of the form b = k ⋯ 21 or b = \overline{ℓ} ⋯ \overline{p}p ⋯ 21 for some 1 ≤ ℓ ≤ p such that 2p − ℓ + 1 = k. For b = k ⋯ 21 everything is determined by weight considerations and it is easy to see that both paths in the diagram (B.2) yield (k − 1) ⋯ 21⊗(k−1) ⋯ 21. For b = \overline{ℓ} ⋯ \overline{p}p ⋯ 21 explicit calculations show that emb_{B}(b) = \overline{ℓ} ⋯ \overline{k}(ℓ − 1) ⋯ 21 ⊗ k ⋯ 21. Hence

− ∘ts(b) = \overline{ℓ + 1} ⋯ \overline{p} p ⋯ 21
emb_{B} ∘ − ∘ ts(b) = \overline{ℓ + 1} ⋯ \overline{k − 1} ℓ ⋯ 21 ⊗ (k − 1) ⋯ 21
− ∘ts ∘ emb_{B}(b) = \overline{ℓ + 1} ⋯ \overline{k} (ℓ − 1) ⋯ 21 ⊗ k ⋯ 21.

From its definition, that R commutes with all crystal operators, one may check that

R(\overline{ℓ + 1} ⋯ \overline{k} (ℓ − 1) ⋯ 21 ⊗ k ⋯ 21) = \overline{ℓ} ⋯ \overline{k − 1} ℓ ⋯ 21 ⊗ (k − 1) ⋯ 21.

Hence indeed emb_{B} ∘ − ∘ ts(b) = − ∘ ts ∘ R ∘ − ∘ ts ∘ emb_{B}(b).

Next we show that if the diagram commutes for b ∈ B^{k,1} then it commutes for f_{i}b for 1 ≤ i ≤ n provided that f_{i}b ≠ 0.

Suppose that b = βb′ where β is a letter in the alphabet A = \{1, 2, …, n, \overline{n}, …, \overline{2}, \overline{1}\} and b′ is a column of height k − 1. We claim that

(B.3)  emb_{B}(b) = βa′ ⊗ a
(B.4)  R(a′ ⊗ a) = βã ⊗ ã′

for some column a′ of height k − 1, a of height k, and columns ã and ã′ of height k − 1. Equations (B.3) and (B.4) are certainly true for classically highest weight elements b as is clear from the previous explicit calculations. Now suppose (B.3) and (B.4) hold for b. Then they hold for f_{i}b for 1 ≤ i ≤ n if f_{i}b ≠ 0 since emb_{B}(B^{k,1}) is aligned and R is a crystal isomorphism so that f_{i}R = Rf_{i}.

If f_{i} does not act on β in b, then emb_{B} ∘ − ∘ ts(f_{i}b) = f_{i}^{2} ∘ emb_{B} ∘ − ∘ ts(b) and

− ∘ts ∘ R ∘ − ∘ ts ∘ emb_{B}(f_{i}b)
= − ∘ts ∘ R ∘ − ∘ ts ∘ f_{i}^{2} ∘ emb_{B}(b)
= − ∘ts ∘ R ∘ f_{i}^{2} ∘ − ∘ ts ∘ emb_{B}(b)
= − ∘ts ∘ f_{i}^{2} ∘ R ∘ − ∘ ts ∘ emb_{B}(b)
=f_{i}^{2} ∘ − ∘ ts ∘ R ∘ − ∘ ts ∘ emb_{B}(b)

by (B.3) and (B.4) so that the diagram still commutes. Similarly, if f_{i} acts on β in b, then emb_{B} ∘ − ∘ ts(f_{i}b) = emb_{B} ∘ − ∘ ts(b) and also

− ∘ts ∘ R ∘ − ∘ ts ∘ emb_{B}(f_{i}b)
= − ∘ts ∘ R ∘ − ∘ ts ∘ f_{i}^{2} ∘ emb_{B}(b)
= − ∘ts ∘ R ∘ f_{i} ∘ − ∘ ts ∘ emb_{B}(b)
= − ∘ts ∘ f_{i} ∘ R ∘ − ∘ ts ∘ emb_{B}(b)
= − ∘ts ∘ R ∘ − ∘ ts ∘ emb_{B}(b)

by (B.3)and (B.4)so that the diagram still commutes. This concludes the proof of Theorem 4.4.

Appendix C. Proof of Lemma 5.5

One may easily verify that (see also [17, Eq. (3.10)])

\begin{matrix}−p_{i−1}^{(a)} + 2p_{i}^{(a)} − p_{i+1}^{(a)} = & − ∑_{b∈J} (α_{a} | α_{b}) m_{i}^{(b)} + δ_{i,1}L^{a} \\ ≥ & − ∑_{b∈J} (α_{a} | α_{b}) m_{i}^{(b)}, \end{matrix} ; (C.1)

The proof of [δ, δ̃] = 0 follows from Lemmas C.1 and C.2 below. The proof of these lemmas is very technical and follows similar arguments as in [17, Appendix A]. Details are available upon request.

Let (ν^{•}, J^{•}) ∈ RC(λ, B) where B = (B^{1,1})^{⊗2} ⊗ B′. The following notation is used:

\begin{aligned}δ(ν^{•}, J^{•}) & = (ν̇^{•}, J̇^{•}) \\ δ̃(ν^{•}, J^{•}) & = (ν̃^{•}, J̃^{•}) \\ δ̃ ∘ δ(ν^{•}, J^{•}) & = (˜ν̇^{•}, ˜ ˙J^{•}) \\ δ ∘ δ̃(ν^{•}, J^{•}) & = ( ˙ν̃^{•}, ˙ J̃^{•}). \end{aligned}

Furthermore, let {ℓ̇^{(k)}, ṡ^{(k)}}, {ℓ̃^{(k)}, s̃^{(k)}}, {˜ℓ̇^{(k)}, ˜ ṡ^{(k)}} and { ˙ℓ̃^{(k)}, ˙ s̃^{(k)}} be the lengths of the strings that are shortened in the transformations (ν^{•}, J^{•}) ↦ (ν̇^{•}, J̇^{•}), (ν^{•}, J^{•}) ↦ (ν̃^{•}, J̃^{•}), (ν̇^{•}, J̇^{•}) ↦ (˜ν̇^{•}, ˜ ˙J^{•}) and (ν̃^{•}, J̃^{•}) ↦ ( ˙ν̃^{•}, ˙ J̃^{•}), respectively. (In Section 4.4 the lengths of the s-strings was denoted by \overline{ℓ}). We call the strings, whose lengths are labeled by an ℓ, ℓ-strings and those labeled by an s, s-strings.

Lemma C.1. The following cases occur at (ν^{•}, J^{•})^{(k)}:

I. Nontwisted case. In this case the ℓ-string selected by δ (resp. δ̃) in (ν^{•}, J^{•})^{(k)} is different from the s-string selected by δ̃ (resp. δ) in (ν^{•}, J^{•})^{(k)}. For the ℓ-strings one of the following must hold: (ℓa) Generic case. If δ and δ̃ do not select the same ℓ-string, then ˙ ℓ̃^{(k)} = ℓ̇^{(k)} and

˜ℓ̇^{(k)} = ℓ̃^{(k)}. (ℓb) Doubly singular case. In this case δ and δ̃ select the same ℓ-string, so that

ℓ̇^{(k)} = ℓ̃^{(k)} =: ℓ. Then

(1) If ˜ ℓ̇^{(k)} < ℓ (or ˙ ℓ̃^{(k)} < ℓ) then ˜ ℓ̇^{(k)} = ˙ ℓ̃^{(k)} = ℓ − 1 and m_{ℓ−1}^{(k+1)} = 0 for k < n − 2, m_{ℓ−1}^{(n−1)} = m_{ℓ−1}^{(n)} = 0 for k = n − 2 and m_{ℓ−1}^{(n−2)} = 0 for k = n − 1, n.

(2) If ˜ ℓ̇^{(k)} = ℓ (or ˙ ℓ̃^{(k)} = ℓ) then case I.(ℓs)(1’) (or I.(ℓs)(1)) holds or ˜ℓ̇^{(k)} = ˙ ℓ̃^{(k)} = ℓ.

(3) If ˜ ℓ̇^{(k)} > ℓ (or ˙ ℓ̃^{(k)} > ℓ) then case I.(ℓs)(1’,2) (or I.(ℓs)(1,2)) holds or ˜ℓ̇^{(k)} = ˙ ℓ̃^{(k)} and ˜ ℓ̇^{(k)} ≤ ℓ̃^{(k+1)}, ˙ ℓ̃^{(k)} ≤ ℓ̇^{(k+1)} for k < n − 2, ˜ ℓ̇^{(n−2)} ≤ min{ℓ̃^{(n−1)}, ℓ̃^{(n)}}, ˙ ℓ̃^{(n−2)} ≤ min{ℓ̇^{(n−1)}, ℓ̇^{(n)}} for k = n − 2, and ˜ℓ̇^{(k)} ≤ s̃^{(n−2)}, ˙ ℓ̃^{(k)} ≤ ṡ^{(n−2)} for k = n − 1, n.

For the s-strings, case I.(ℓs) holds or one the following must hold: (sa) Generic case. If δ and δ̃ do not select the same s-string, then ˙ s̃^{(k)} = ṡ^{(k)} and

˜ṡ^{(k)} = s̃^{(k)}. (sb) Doubly singular case. In this case δ and δ̃ select the same s-string, so that

ṡ^{(k)} = s̃^{(k)} =: s. Then (1) If ˜ ṡ^{(k)} < s (or ˙ s̃^{(k)} < s) then ˜ ṡ^{(k)} = ˙ s̃^{(k)} = s − 1 and m_{s−1}^{(k−1)} = 0.

(2) If ˜ ṡ^{(k)} = s (or ˙ s̃^{(k)} = s) then ˜ ṡ^{(k)} = ˙ s̃^{(k)} = s.

(3) If ˜ ṡ^{(k)} > s (or ˙ s̃^{(k)} > s) then ˜ ṡ^{(k)} = ˙ s̃^{(k)}, ˜ ṡ^{(k)} ≤ s̃^{(k−1)} and ˙ s̃^{(k)} ≤

ṡ^{(k−1)}.

(ℓs) Mixed case. One of the following holds:

(1) ℓ̇^{(k)} = ℓ̃^{(k)} =: ℓ, ˙ ℓ̃^{(k)} = ṡ^{(k)} = ℓ̇^{(k+1)} =: ℓ′, ˜ ℓ̇^{(k)} = ˙ s̃^{(k)} =: ℓ′′, s̃^{(k)} = ˜ ṡ^{(k)} or possibly the same conditions for ℓ and ℓ′, ˜ ℓ̇^{(k)} = s̃^{(k)} = s̃^{(k+1)} = ℓ′′, ˙ s̃^{(k)} = ˜ ṡ^{(k)} = ℓ′′′, m_{ℓ′′}^{(k−1)} = 0, m_{ℓ′′}^{(k)} = 1, m_{ℓ′′}^{(k+1)} = 2 if

case I.(ℓs)(1) does not hold at k −1. Furthermore, either ˜ ℓ̇^{(k)} ≤ ℓ̃^{(k+1)} or case I.(ℓs)(1) holds at k + 1 with the same values of ℓ′ and ℓ′′. Similarly, either ˙ s̃^{(k)} ≤ ṡ^{(k−1)} or case I.(ℓs)(1) holds at k − 1 with the same values of ℓ′ and ℓ′′.

(1’) ℓ̇^{(k)} = ℓ̃^{(k)} =: ℓ, ˜ ℓ̇^{(k)} = s̃^{(k)} = ℓ̃^{(k+1)} =: ℓ′, ˙ ℓ̃^{(k)} = ˜ ṡ^{(k)} =: ℓ′′, ṡ^{(k)} = ˙ s̃^{(k)} or possibly the same conditions for ℓ and ℓ′, ˙ ℓ̃^{(k)} = ṡ^{(k)} = ṡ^{(k+1)} = ℓ′′, ˜ ṡ^{(k)} = ˙ s̃^{(k)} = ℓ′′′, m_{ℓ′′}^{(k−1)} = 0, m_{ℓ′′}^{(k)} = 1, m_{ℓ′′}^{(k+1)} = 2 if

case I.(ℓs)(1’) does not hold at k−1. Furthermore, either ˙ ℓ̃^{(k)} ≤ ℓ̇^{(k+1)} or case I.(ℓs)(1’) holds at k + 1 with the same values of ℓ′ and ℓ′′. Similarly, either ˜ ṡ^{(k)} ≤ s̃^{(k−1)} or case I.(ℓs)(1’) holds at k − 1 with the same values of ℓ′ and ℓ′′.

(2) For k < n − 2 (resp. k = n − 2) ℓ̇^{(k)} = ℓ̃^{(k)} =: ℓ, ṡ^{(k)} = s̃^{(k)} = ṡ^{(k+1)} = s̃^{(k+1)} =: ℓ′ (resp. ṡ^{(k)} = s̃^{(k)} = ℓ̇^{(n−1)} = ℓ̇^{(n)} = ℓ̃^{(n−1)} = ℓ̃^{(n)} = ℓ′), ˙ ℓ̃^{(k)} = ˜ ℓ̇^{(k)} = ℓ′′, ˙ s̃^{(k)} = ˜ ṡ^{(k)} := ℓ′′′ and case I.(ℓs)(2) holds at k + 1 (resp. n − 1 and n) with the same values of ℓ′ and ℓ′′ and ℓ = ℓ′, ℓ′′′ = ℓ′′. Also, either ˜ ṡ^{(k)} ≤ s̃^{(k−1)} and ˙ s̃^{(k)} ≤ ṡ^{(k−1)} or case I.(ℓs)(2) holds at k − 1 with the same values of ℓ′ and ℓ′′. For k = n − 1, n, ℓ̇^{(k)} = ℓ̃^{(k)} = ṡ^{(n−2)} = s̃^{(n−2)} = ℓ′ and ˜ ℓ̇^{(k)} = ˙ ℓ̃^{(k)} = ℓ′′. In addition case I.(ℓs)(2) holds at n − 2 with the same values of ℓ′ and ℓ′′.

II. Twisted case. In this case the ℓ-string in (ν^{•}, J^{•})^{(k)} selected by δ is the same as the s-string selected by δ̃ or vice versa. In the first case ℓ̇^{(k)} = s̃^{(k)} =: ℓ. Then ℓ̃^{(k)} = ˜ ℓ̇^{(k)} and one of the following holds:

(1) If ˙ ℓ̃^{(k)} < ℓ, then ˙ ℓ̃^{(k)} = ˜ ṡ^{(k)} = ℓ − 1, m_{ℓ−1}^{(k+1)} = 0 or m_{ℓ−1}^{(k+1)}(ν̃^{•}) = 0, and m_{ℓ−1}^{(k−1)} = 0 or m_{ℓ−1}^{(k−1)}(ν̇^{•}) = 0. Furthermore ṡ^{(k)} = ˙ s̃^{(k)}.

(2) If ˙ ℓ̃^{(k)} = ℓ, then ˙ ℓ̃^{(k)} = ˜ ṡ^{(k)} = ℓ and ṡ^{(k)} = ˙ s̃^{(k)}.

(3) If ˙ ℓ̃^{(k)} > ℓ, then (i) ˙ ℓ̃^{(k)} = ˜ ṡ^{(k)} and ṡ^{(k)} = ˙ s̃^{(k)}, or (ii) ˙ ℓ̃^{(k)} = ṡ^{(k)} and ˜ ṡ^{(k)} = ˙ s̃^{(k)} ≤ ṡ^{(k−1)}. Furthermore, either ˙ ℓ̃^{(k)} ≤ ℓ̇^{(k+1)} or ℓ̇^{(k)} = ℓ̇^{(k+1)}, ˙ ℓ̃^{(k)} = ˙ ℓ̃^{(k+1)}, m_{ℓ}^{(k+1)} = 1 and Case II.(3)(i) holds at k + 1. Similarly, either ˜ ṡ^{(k)} ≤ s̃^{(k−1)} or ℓ̇^{(k)} = ℓ̇^{(k−1)}, ˙ ℓ̃^{(k)} = ˙ ℓ̃^{(k−1)}, m_{ℓ}^{(k−1)} = 1 and Case II.(3) holds at k − 1.

If the ℓ-string in (ν^{•}, J^{•})^{(k)} selected by δ̃ is the same as the s-string selected by δ, then ℓ̃^{(k)} = ṡ^{(k)} =: ℓ. In this case ℓ̇^{(k)} = ˙ ℓ̃^{(k)} and one of the following holds: (1’) If ˜ ℓ̇^{(k)} < ℓ, then ˜ ℓ̇^{(k)} = ˙ s̃^{(k)} = ℓ − 1, m_{ℓ−1}^{(k+1)} = 0 or m_{ℓ−1}^{(k+1)}(ν̇^{•}) = 0, and

m_{ℓ−1}^{(k−1)} = 0 or m_{ℓ−1}^{(k−1)}(ν̃^{•}) = 0. Furthermore s̃^{(k)} = ˜ ṡ^{(k)}.

(2’) If ˜ ℓ̇^{(k)} = ℓ, then ˜ ℓ̇^{(k)} = ˙ s̃^{(k)} = ℓ and s̃^{(k)} = ˜ ṡ^{(k)}. (3’) If ˜ ℓ̇^{(k)} > ℓ, then

(i) ˜ ℓ̇^{(k)} = ˙ s̃^{(k)} and s̃^{(k)} = ˜ ṡ^{(k)}, or (ii) ˜ ℓ̇^{(k)} = s̃^{(k)} and ˙ s̃^{(k)} = ˜ ṡ^{(k)} ≤ ℓ̃^{(k−1)}. Furthermore, either ˜ ℓ̇^{(k)} ≤ ℓ̃^{(k+1)} or ℓ̃^{(k)} = ℓ̃^{(k+1)}, ˜ ℓ̇^{(k)} = ˜ ℓ̇^{(k+1)}, m_{ℓ}^{(k+1)} = 1 and Case II.(3’)(i) holds at k + 1. Similarly, either ˙ s̃^{(k)} ≤ ṡ^{(k−1)} or ℓ̃^{(k)} = ℓ̃^{(k−1)}, ˜ ℓ̇^{(k)} = ˜ ℓ̇^{(k−1)}, m_{ℓ}^{(k−1)} = 1 and Case II.(3’) holds at k − 1.

Lemma C.2. ˜ ˙J^{•} = ˙ J̃^{•}.

Appendix D. Combinatorial R-matrix on highest weight elements

Here we give the combinatorial R-matrix R : B^{k′,1} ⊗ B^{k,1} → B^{k,1} ⊗ B^{k′,1} explicitly for highest weight elements. The following notation is employed. A column with entries ⋯ a_{4} ⋯ a_{3}a_{2} ⋯ a_{1}, where a_{2} ⋯ a_{1} denotes a sequence of consecutive integers a_{2} a_{2} − 1 a_{2}−2 … a_{1}+1 a_{1}, is compactly written as [a_{1}..a_{2} | a_{3}..a_{4} | ⋯]. We use the convention that a segment [.. | a..b | ..] is empty if b < a.

Since R is an involution, it suffices to state R for the case k′ ≤ k. For k′ = k we have R = id. Hence for R : B^{n,1} ⊗ B^{n,1} → B^{n,1} ⊗ B^{n,1} the R-matrix is the identity. The highest weight elements are given by [1..n − 2ℓ | \overline{n}..\overline{n − 2ℓ + 1}] ⊗ [1..n] for 0 ≤ ℓ ≤ n/2 with local energy H = ℓ. The same results hold for R : B^{n−1,1} ⊗ B^{n−1,1} → B^{n−1,1} ⊗ B^{n−1,1} with n and \overline{n} interchanged.

For R : B^{n−1,1} ⊗ B^{n,1} → B^{n,1} ⊗ B^{n−1,1} the highest weight elements get mapped as follows:

\begin{matrix}[1..n − 2ℓ − 1 | \overline{n}..\overline{n − 2ℓ}] ⊗ [1..n] \\ ⟷^{R} & [1..n − 2ℓ − 1 | n | \overline{n − 1}..\overline{n − 2ℓ}] ⊗ [1..n − 1 | \overline{n}] \end{matrix}

where 0 ≤ ℓ < n/2. The local energy is H = ℓ.

Next consider the case R : B^{k′,1} ⊗ B^{n,1} → B^{n,1} ⊗ B^{k′,1} where k′ ≤ n − 2. The highest weight elements are [1..ℓ | \overline{n}n...\overline{n}n | \overline{ℓ}..\overline{r}] ⊗ [1..n] and [1..ℓ | \overline{n}n...\overline{n} | \overline{ℓ}..\overline{r}] ⊗ [1..n] where r ≤ ℓ + 1 and the pieces \overline{n}n..\overline{n}n and \overline{n}n..\overline{n} could have length zero. Then

\begin{matrix}[1..ℓ | \overline{n}n...\overline{n}n | \overline{ℓ}..\overline{r}] ⊗ [1..n] ⟷^{R} & [1..r − 1 | ℓ + d + 1..n | \overline{ℓ + d}..\overline{r}] \\ ⊗ & [1..\frac{k′ + ℓ + d}{2} | \overline{\frac{k′ + ℓ + d}{2}}..\overline{ℓ + d + 1}] \end{matrix}

with H = \frac{k′−ℓ+d}{2} and

\begin{matrix}[1..ℓ | \overline{n}n...\overline{n} | \overline{ℓ}..\overline{r}] ⊗ [1..n] ⟷^{R} & [1..r − 1 | ℓ + d + 2..n − 1 | \overline{n} | \overline{ℓ + d + 1}..\overline{r}] \\ ⊗ & [1..\frac{k′ + ℓ + d + 1}{2} | \overline{\frac{k′ + ℓ + d + 1}{2}}..\overline{ℓ + d + 2}] \end{matrix}

with H = \frac{k′−ℓ+d+1}{2}.

Similarly, the case R : B^{k′,1} ⊗ B^{n−1,1} → B^{n−1,1} ⊗ B^{k′,1} with k′ ≤ n − 2 is obtained from the above by interchanging n and \overline{n} everywhere.

The remaining cases are all of the form R : B^{k′,1} ⊗ B^{k,1} → B^{k,1} ⊗ B^{k′,1} with 1 ≤ k′ ≤ k ≤ n − 2. The highest weight elements u′ ⊗ u ∈ B^{k′,1} ⊗ B^{k,1} are summarized in the following table:

Case u′ ⊗ u  parameter ranges
1 [1..ℓ | k + 1..p | \overline{p}..\overline{q} | \overline{ℓ}..\overline{r}] ⊗ [1..k] ℓ ≥ 0; p ≥ k; q ≤ p + 1; r ≤ ℓ + 1
2a [1..ℓ | \overline{ℓ}..\overline{r}] ⊗ [1..a | \overline{a}..\overline{b}]  a ≥ b; ℓ ≥ 1; r ≤ ℓ + 1
2b [1..ℓ | \overline{ℓ}..\overline{r} | \overline{b − 1}..\overline{q}] ⊗ [1..a | \overline{a}..\overline{b}] a ≥ b; ℓ ≥ r > b > q ≥ 1
3 [1..ℓ | b..p | \overline{ℓ}..\overline{r}] ⊗ [1..a | \overline{a}..\overline{b}]  a ≥ b; ℓ ≥ 0; p ≥ b > ℓ + 1 ≥ r
4 [1..ℓ | b..p | \overline{p}..\overline{q} | \overline{ℓ}..\overline{r}] ⊗ [1..a | \overline{a}..\overline{b}] a ≥ b; ℓ ≥ 0; p ≥ b, q; r ≤ ℓ + 1
5a [1..ℓ | \overline{b − 1}..\overline{q}] ⊗ [1..a | \overline{a}..\overline{b}]  a ≥ b > q; ℓ ≥ 0
5b [1..ℓ | \overline{b − 1}..\overline{q} | \overline{ℓ}..\overline{r}] ⊗ [1..a | \overline{a}..\overline{b}] a ≥ b > q > ℓ + 1; ℓ ≥ r ≥ 1

In cases 1 and 4 ..p | \overline{p}.. may also be replaced by ..n\overline{n} ⋯ n\overline{n}.. or ..\overline{n}n ⋯ \overline{n}n...

The answers for R(u′ ⊗ u) in the various cases are given in the tables below. For brevity, we use the definition d = ℓ−r +1. By weight considerations R([1..k′]⊗[1..k]) = [1..k] ⊗ [1..k′]. All answers are proven by induction. As an example, we demonstrate how to derive Case 2a with r ≤ ℓ and k′ < b ≤ a − d by induction on d. We assume that Case 2a holds for d = 0 (which can also be proven by induction).

First consider d = 1 so that u = [1..ℓ | \overline{ℓ}] ⊗ [1..a | \overline{a}..\overline{b}]. Then k′ < b implies that ℓ < b − 1 so that

\begin{aligned}u′ & = f_{1}f_{2} ⋯ f_{ℓ−1}(u) = [1..ℓ | \overline{1}] ⊗ [1..a | \overline{a}..\overline{b}] \\ u′′ & = f_{2}f_{3} ⋯ f_{ℓ}(u′) = [1 | 3..ℓ + 1 | \overline{1}] ⊗ [1..a | \overline{a}..\overline{b}] \\ u′′′ & = f_{0}(u′′) = [1..ℓ + 1] ⊗ [1..a | \overline{a}..\overline{b}]. \end{aligned}

By induction hypothesis, v′′′ = R(u′′′) = [1..a | \overline{a}..\overline{b}] ⊗ [1..ℓ + 1] and

\begin{aligned}v′′ & = e_{0}(v′′′) = [1..a − 1 | \overline{a − 1}..\overline{b} | \overline{2}\overline{1}] ⊗ [1..ℓ + 1] \\ v′ & = e_{ℓ}e_{ℓ−1} ⋯ e_{2}(v′′) = [1..a − 1 | \overline{a − 1}..\overline{b} | \overline{ℓ + 1} | \overline{1}] ⊗ [1..ℓ + 1] \\ v & = e_{ℓ−1}e_{ℓ−2} ⋯ e_{1}(v′) = [1..a − 1 | \overline{a − 1}..\overline{b} | \overline{ℓ + 1} \overline{ℓ}] ⊗ [1..ℓ + 1] \end{aligned}

which by (2.7) and (2.1) indeed yields v = R(u) = [1..a − d | \overline{a − d}..\overline{b} | \overline{k′}..\overline{r}] ⊗ [1..k′]. Note also that H(u′′′) = 0 by induction hypothesis. Since f_{0} acts on the left tensor factor in both u′′ and v′′, we have H(u) = 1.

Next consider d > 1 so that u = [1..ℓ | \overline{ℓ}..\overline{r}] ⊗ [1..a | \overline{a}..\overline{b}] with r < ℓ. Then r ≤ ℓ − 1 ≤ k′ − 2 < b − 2 so that

\begin{aligned}u′ & = f_{2}f_{3} ⋯ f_{r}f_{1}f_{2} ⋯ f_{r−1}(u) = [1..ℓ | \overline{ℓ}..\overline{r + 2} | \overline{2} \overline{1}] ⊗ [1..a | \overline{a}..\overline{b}] \\ u′′ & = f_{0}(u′) = [1..ℓ + 1 | \overline{ℓ + 1}..\overline{r + 2}] ⊗ [1..a | \overline{a}..\overline{b}]. \end{aligned}

By induction hypothesis, v′′ = R(u′′) = [1..a − d + 1 | \overline{a − d + 1}..\overline{b} | \overline{k′}..\overline{r + 2}] ⊗ [1..k′] and

\begin{aligned}v′ & = e_{0}(v′′) = [1..a − d | \overline{a − d}..\overline{b} | \overline{k′}..\overline{r + 2} | \overline{2} \overline{1}] ⊗ [1..k′] \\ v & = e_{r−1}e_{r−2} ⋯ e_{1}e_{r}e_{r−1} ⋯ e_{2}(v′) = [1..a − d | \overline{a − d}..\overline{b} | \overline{k′}..\overline{r}] ⊗ [1..k′] \end{aligned}

which yields v = R(u) = [1..a − d | \overline{a − d}..\overline{b} | \overline{k′}..\overline{r}] ⊗ [1..k′] by (2.7) and (2.1). By induction hypothesis H(u′′) = k′ − ℓ − 1. Since f_{0} acts on the left tensor factor in both u′ and v′, we have H(u) = k′ − ℓ as claimed.

Case R(u′ ⊗ u)  conditions  H(u′ ⊗ u)
1 [1..ℓ + 1 | k′ + 1..k − 1 | \overline{k − 1}..\overline{q} | \overline{ℓ + 1}..\overline{r}] ⊗ [1..k′]  p = k and k − q even k′ − ℓ
[1..ℓ | k′ + 1..p | \overline{p}..\overline{q} | \overline{ℓ}..\overline{r}] ⊗ [1..k′]  otherwise  k′ − ℓ
2a [1..a − d | \overline{a − d}..\overline{b} | \overline{k′}..\overline{r}] ⊗ [1..k′]  k′ < b ≤ a − d  k′ − ℓ
[1..a − b + ℓ + 1 | k′ + 1..b − 1 | \overline{a − b + ℓ + 1}..\overline{r}] ⊗ [1..k′]  k′ < b; a − d < b  k′ − ℓ
[1..\frac{k−k′+2ℓ}{2} | \overline{\frac{k−k′+2ℓ}{2}}..\overline{r}] ⊗ [1..\frac{k′+b−1}{2} | \overline{\frac{k′+b−1}{2}}..\overline{b}]  b ≤ k′, k + k′ even  k′ − \frac{b+r−2}{2}
[1..b − 1 | b + 1..\frac{k−k′+2ℓ+1}{2} | \overline{\frac{k−k′+2ℓ+1}{2}}..\overline{r}] ⊗ [1..\frac{k′+b}{2} | \overline{\frac{k′+b}{2}}..\overline{b + 1}]  b ≤ k′, k + k′ odd  k′ − \frac{b+r−3}{2}
2b [1..\frac{k−k′+2ℓ}{2} | \overline{\frac{k−k′+2ℓ}{2}}..\overline{r} | \overline{b − 1}..\overline{q}] ⊗ [1..\frac{k′+b−1}{2} | \overline{\frac{k′+b−1}{2}}..\overline{b}]  k + k′ even  k′ − \frac{r+q−2}{2}
[1..b − 1 | b + 1..\frac{k−k′+2ℓ+1}{2} | \overline{\frac{k−k′+2ℓ+1}{2}}..\overline{r} | \overline{b − 1}..\overline{q}] ⊗ [1..\frac{k′+b}{2} | \overline{\frac{k′+b}{2}}..\overline{b}]  k + k′ odd  k′ − \frac{r+q−3}{2}
3 [1..ℓ | k′ − k + b..p | \overline{ℓ}..\overline{r}] ⊗ [1..k′ − k + a | \overline{k′ − k + a}..\overline{k′ − k + b}]  k < p  k′ − k + a − ℓ
[1..\frac{k+p−2d}{2} | \overline{\frac{k+p−2d}{2}}..\overline{p + 1} | \overline{ℓ + d}..\overline{r}] ⊗ [1..\frac{k′+ℓ+d}{2} | \overline{\frac{k′+ℓ+d}{2}}..\overline{ℓ + d + 1}]  k′ ≤ p ≤ k, k + p even \frac{k′−r+1}{2}
[1..ℓ + d | ℓ + d + 2..\frac{k+p−2d+1}{2} | \overline{\frac{k+p−2d+1}{2}}..\overline{p + 1} | \overline{ℓ + d}..\overline{r}]  k′ ≤ p ≤ k, k + p odd \frac{k′−r+2}{2}

⊗[1..\frac{k′+ℓ+d+1}{2} | \overline{\frac{k′+ℓ+d+1}{2}}..\overline{ℓ + d + 2}]

[1..\frac{k+k′−2d}{2} | \overline{\frac{k+k′−2d}{2}}..\overline{p + 1} | \overline{b − 1}..\overline{r}] ⊗ [1..\frac{k′+b−1}{2} | \overline{\frac{k′+b−1}{2}}..\overline{b}]  p < k′, k + k′ even  \frac{k′−b+2d+1}{2}
[1..b − 1 | b + 1..\frac{k+k′−2d+1}{2} | \overline{\frac{k+k′−2d+1}{2}}..\overline{p + 1} | \overline{b − 1}..\overline{r}] ⊗ [1..\frac{k′+b}{2} | \overline{\frac{k′+b}{2}}..\overline{b + 1}] p < k′, k + k′ odd  \frac{k′−b+2d+2}{2}

33

4 [1..ℓ | k′ − k + b..p | \overline{p}..\overline{q} | \overline{ℓ}..\overline{r}] ⊗ [1..k′ − k + a | \overline{k′ − k + a}..\overline{k′ − k + b}] k < p  k′ − k + a − ℓ
[1..\frac{2ℓ+k−p+2}{2} | k′ + b − p..p − 1 | \overline{p − 1}..\overline{q} | \overline{\frac{2ℓ+k−p+2}{2}}..\overline{r}]  k − 2d < p ≤ k, k′ < p k′ − ℓ + \frac{b−p−1}{2}

⊗[1..\frac{2k′+b−p−1}{2} | \overline{\frac{2k′+b−p−1}{2}}..\overline{k′ + b − p}] k + p even, p − q even

[1..\frac{2ℓ+k−p+1}{2} | k′ + b − p + 1..p | \overline{p}..\overline{q} | \overline{\frac{2ℓ+k−p+1}{2}}..\overline{r}]  k − 2d < p ≤ k, k′ < p k′ − ℓ + \frac{b−p}{2}
⊗[1..\frac{2k′+b−p}{2} | \overline{\frac{2k′+b−p}{2}}..\overline{k′ + b − p + 1}]  k + p odd, p − q even
[1..\frac{2ℓ+k−p}{2} | k′ + b − p..p | \overline{p}..\overline{q} | \overline{\frac{2ℓ+k−p}{2}}..\overline{r}]  k − 2d < p ≤ k, k′ < p k′ − ℓ + \frac{b−p−1}{2}
⊗[1..\frac{2k′+b−p−1}{2} | \overline{\frac{2k′+b−p−1}{2}}..\overline{k′ + b − p}]  k + p even, p − q odd
[1..\frac{2ℓ+k−p−1}{2} | k′ + b − p + 1..p + 1 | \overline{p + 1}..\overline{q} | \overline{\frac{2ℓ+k−p−1}{2}}..\overline{r}]  k − 2d < p ≤ k, k′ < p k′ − ℓ + \frac{b−p}{2}
⊗[1..\frac{2k′+b−p}{2} | \overline{\frac{2k′+b−p}{2}}..\overline{k′ + b − p + 1}]  k + p odd, p − q odd
[1..ℓ + 1 + d | k′ + b − p..\frac{k+p−2d−2}{2} | \overline{\frac{k+p−2d−2}{2}}..\overline{q} | \overline{ℓ + 1 + d}..\overline{r}]  k′ < p ≤ k − 2d  k′ − ℓ + \frac{b−p−1}{2}

⊗[1..\frac{2k′+b−p−1}{2} | \overline{\frac{2k′+b−p−1}{2}}..\overline{k′ + b − p}] k + p even, p − q even

[1..ℓ + 1 + d | k′ + b − p + 1..\frac{k+p−2d−1}{2} | \overline{\frac{k+p−2d−1}{2}}..\overline{q} | \overline{ℓ + 1 + d}..\overline{r}]  k′ < p ≤ k − 2d  k′ − ℓ + \frac{b−p}{2}
⊗[1..\frac{2k′+b−p}{2} | \overline{\frac{2k′+b−p}{2}}..\overline{k′ + b − p + 1}]  k + p odd, p − q even
[1..ℓ + d | k′ + b − p..\frac{k+p−2d}{2} | \overline{\frac{k+p−2d}{2}}..\overline{q} | \overline{ℓ + d}..\overline{r}]  k′ < p ≤ k − 2d  k′ − ℓ + \frac{b−p−1}{2}
⊗[1..\frac{2k′+b−p−1}{2} | \overline{\frac{2k′+b−p−1}{2}}..\overline{k′ + b − p}]  k + p even, p − q odd
[1..ℓ + d | k′ + b − p + 1..\frac{k+p−2d+1}{2} | \overline{\frac{k+p−2d+1}{2}}..\overline{q} | \overline{ℓ + d}..\overline{r}]  k′ < p ≤ k − 2d  k′ − ℓ + \frac{b−p}{2}
⊗[1..\frac{2k′+b−p}{2} | \overline{\frac{2k′+b−p}{2}}..\overline{k′ + b − p + 1}]  k + p odd, p − q odd

34

4 [1..\frac{2ℓ+k−k′}{2} | b..p | \overline{p}..\overline{q} | \overline{\frac{2ℓ+k−k′}{2}} ..\overline{r}] ⊗ [1..\frac{k′+b−1}{2} | \overline{\frac{k′+b−1}{2}}..\overline{b}]  \frac{k+k′−2d}{2} < p ≤ k′, k + k′ even \begin{cases}\frac{2k′−r−q+4}{2} & p − q even \\ \frac{2k′−r−q+2}{2} & p − q odd \end{cases}
cont. [1..\frac{2ℓ+k−k′+1}{2} | b − 1..p − 1 | \overline{p − 1}..\overline{q} | \overline{\frac{2ℓ+k−k′+1}{2}}..\overline{r}]  \frac{k+k′−2d+1}{2} < p ≤ k′  \frac{2k′−r−q+3}{2}
⊗[1..\frac{k′+b−2}{2} | \overline{\frac{k′+b−2}{2}}..\overline{b − 1}]  k + k′ odd, p − q even
[1..\frac{2ℓ+k−k′−1}{2} | b + 1..p + 1 | \overline{p + 1}..\overline{q} | \overline{\frac{2ℓ+k−k′−1}{2}}..\overline{r}]  \frac{k+k′−2d−1}{2} < p ≤ k′  \frac{2k′−r−q+3}{2}
⊗[1..\frac{k′+b}{2} | \overline{\frac{k′+b}{2}}..\overline{b + 1}]  k + k′ odd, p − q odd
[1..ℓ + p − k′ + d | b..\frac{k+k′−2d}{2} | \overline{\frac{k+k′−2d}{2}}..\overline{q} | \overline{ℓ + p − k′ + d}..\overline{r}]  k′ < p + d ≤ \frac{k+k′}{2} , p ≤ k′  \begin{cases}\frac{2k′−r−q+4}{2} & p − q even \\ \frac{2k′−r−q+2}{2} & p − q odd \end{cases}
⊗[1..\frac{k′+b−1}{2} | \overline{\frac{k′+b−1}{2}}..\overline{b}]  k + k′ even
[1..ℓ + p − k′ + d | b − 1..\frac{k+k′−2d−1}{2} | \overline{\frac{k+k′−2d−1}{2}}..\overline{q} | \overline{ℓ + p − k′ + d}..\overline{r}] k′ < p + d ≤ \frac{k+k′+1}{2}, p ≤ k′  \frac{2k′−r−q+3}{2}
⊗[1..\frac{k′+b−2}{2} | \overline{\frac{k′+b−2}{2}}..\overline{b − 1}]  k + k′ odd, p − q even
[1..ℓ + p − k′ + d | b + 1..\frac{k+k′−2d+1}{2} | \overline{\frac{k+k′−2d+1}{2}}..\overline{q} | \overline{ℓ + p − k′ + d}..\overline{r}] k′ < p + d ≤ \frac{k+k′−1}{2}, p ≤ k′  \frac{2k′−r−q+3}{2}
⊗[1..\frac{k′+b}{2} | \overline{\frac{k′+b}{2}}..\overline{b + 1}]  k + k′ odd, p − q odd
[1..ℓ | b..\frac{k+2p−k′}{2} | \overline{\frac{k+2p−k′}{2}} ..\overline{q} | \overline{ℓ}..\overline{r}] ⊗ [1..\frac{k′+b−1}{2} | \overline{\frac{k′+b−1}{2}}..\overline{b}]  p ≤ k′ − d, k + k′ even  \begin{cases}\frac{2k′−r−q+4}{2} & b − ℓ even \\ \frac{2k′−r−q+2}{2} & b − ℓ odd \end{cases}
[1..ℓ | b − 1..\frac{k+2p−k′−1}{2} | \overline{\frac{k+2p−k′−1}{2}}..\overline{q} | \overline{ℓ}..\overline{r}]  p ≤ k′ − d  \frac{2k′−r−q+3}{2}
⊗[1..\frac{k′+b−2}{2} | \overline{\frac{k′+b−2}{2}}..\overline{b − 1}]  k + k′ odd, b − ℓ even
[1..ℓ | b + 1..\frac{k+2p−k′+1}{2} | \overline{\frac{k+2p−k′+1}{2}}..\overline{q} | \overline{ℓ}..\overline{r}]  p ≤ k′ − d  \frac{2k′−r−q+3}{2}
⊗[1..\frac{k′+b}{2} | \overline{\frac{k′+b}{2}}..\overline{b + 1}]  k + k′ odd, b − ℓ odd

35

5a [1..ℓ | k′ + 1..a | \overline{a}..\overline{q}] ⊗ [1..k′]  q > ℓ + 1, k′ − ℓ even  k′ − ℓ
[1..ℓ + 1 | k′ + 1..a − 1 | \overline{a − 1}..\overline{q} | \overline{ℓ + 1}] ⊗ [1..k′]  q > ℓ + 1, k′ − ℓ odd  k′ − ℓ
[1..ℓ | b..\frac{k+2b−k′−2}{2} | \overline{\frac{k+2b−k′−2}{2}}..\overline{q}] ⊗ [1..\frac{k′+b−1}{2} | \overline{\frac{k′+b−1}{2}}..\overline{b}]  q ≤ ℓ + 1 < b, k + k′ even \begin{cases}\frac{2k′−ℓ−q+3}{2} & b − ℓ even \\ \frac{2k′−ℓ−q+1}{2} & b − ℓ odd \end{cases}
[1..ℓ | b − 1..\frac{k+2b−k′−3}{2} | \overline{\frac{k+2b−k′−3}{2}}..\overline{q}] ⊗ [1..\frac{k′+b−2}{2} | \overline{\frac{k′+b−2}{2}}..\overline{b − 1}] q ≤ ℓ + 1 < b, k + k′ odd \frac{2k′−ℓ−q+2}{2}

b − ℓ even

[1..ℓ | b + 1..\frac{k+2b−k′−1}{2} | \overline{\frac{k+2b−k′−1}{2}}..\overline{q}] ⊗ [1..\frac{k′+b}{2} | \overline{\frac{k′+b}{2}}..\overline{b + 1}]  q ≤ ℓ + 1 < b, k + k′ odd \frac{2k′−ℓ−q+2}{2}

b − ℓ odd

[1..\frac{k+2ℓ−k′}{2} | \overline{\frac{k+2ℓ−k′}{2}} ..\overline{ℓ + 1} | \overline{b − 1}..\overline{q}] ⊗ [1..\frac{k′+b−1}{2} | \overline{\frac{k′+b−1}{2}}..\overline{b}]  q, b ≤ ℓ + 1, k + k′ even \frac{k′+b−2q+1}{2}
[1..b − 1 | b + 1..\frac{k+2ℓ−k′+1}{2} | \overline{\frac{k+2ℓ−k′+1}{2}}..\overline{ℓ + 1} | \overline{b − 1}..\overline{q}]  q, b ≤ ℓ + 1, k + k′ odd  \frac{k′+b−2q+2}{2}
⊗[1..\frac{k′+b}{2} | \overline{\frac{k′+b}{2}}..\overline{b + 1}]

36

5b [1..ℓ + d + 1 | k′ + 1..a − 1 − d | \overline{a − 1 − d}..\overline{q} | \overline{ℓ + d + 1}..\overline{r}]  q ≥ ℓ + d + 2, k′ − r even  k′ − ℓ
⊗[1..k′]  a − b ≥ d − 1
[1..ℓ + d | k′ + 1..a − d | \overline{a − d}..\overline{q} | \overline{ℓ + d}..\overline{r}]  q ≥ ℓ + d + 2, k′ − r odd  k′ − ℓ
⊗[1..k′]  a − b ≥ d − 1
[1..ℓ + a − b + 2 | k′ + 1..b − 2 | \overline{b − 2}..\overline{q} | \overline{ℓ + a − b + 2}..\overline{r}]  q ≥ ℓ + d + 2, k′ − r even  k′ − ℓ
⊗[1..k′]  a − b < d − 1
[1..ℓ + a − b + 1 | k′ + 1..b − 1 | \overline{b − 1}..\overline{q} | \overline{ℓ + a − b + 1}..\overline{r}]  q ≥ ℓ + d + 2, k′ − r odd  k′ − ℓ
⊗[1..k′]  a − b < d − 1
[1..q − 1 | b..a − \frac{q−r}{2} | \overline{a − \frac{q−r}{2}} ..\overline{r}] ⊗ [1..\frac{k′+b−1}{2} | \overline{\frac{k′+b−1}{2}}..\overline{b}]  ℓ + d + 1 ≥ q > ℓ + 1  \begin{cases}\frac{2k′−r−q+4}{2} & k′ − r even \\ \frac{2k′−r−q+2}{2} & k′ − r odd \end{cases}

r − q even, a − b ≥ \frac{q−r−2}{2}

[1..a − b + \frac{q+r}{2} | \overline{b − 1}..\overline{q} | \overline{a − b + \frac{q+r}{2}} ..\overline{r}] ⊗ [1..\frac{k′+b−1}{2} | \overline{\frac{k′+b−1}{2}}..\overline{b}] ℓ + d + 1 ≥ q > ℓ + 1  \begin{cases}\frac{2k′−r−q+4}{2} & k′ − r even \\ \frac{2k′−r−q+2}{2} & k′ − r odd \end{cases}

r − q even, a − b < \frac{q−r−2}{2}

[1..q − 1 | b − 1..a − \frac{q−r+1}{2} | \overline{a − \frac{q−r+1}{2}}..\overline{r}]  ℓ + d + 1 ≥ q > ℓ + 1, r − q odd \frac{2k′−r−q+3}{2}
⊗[1..\frac{k′+b−2}{2} | \overline{\frac{k′+b−2}{2}}..\overline{b − 1}]  k′ − r even, a − b ≥ \frac{q−r−3}{2}
[1..a − b + \frac{q+r+1}{2} | \overline{b − 2}..\overline{q} | \overline{a − b + \frac{q+r+1}{2}}..\overline{r}]  ℓ + d + 1 ≥ q > ℓ + 1, r − q odd \frac{2k′−r−q+3}{2}
⊗[1..\frac{k′+b−2}{2} | \overline{\frac{k′+b−2}{2}}..\overline{b − 1}]  k′ − r even, a − b < \frac{q−r−3}{2}
[1..q − 1 | b + 1..a − \frac{q−r−1}{2} | \overline{a − \frac{q−r−1}{2}}..\overline{r}]  ℓ + d + 1 ≥ q > ℓ + 1, r − q odd \frac{2k′−r−q+3}{2}
⊗[1..\frac{k′+b}{2} | \overline{\frac{k′+b}{2}}..\overline{b + 1}]  k′ − r odd, a − b ≥ \frac{q−r−1}{2}
[1..a − b + \frac{q+r−1}{2} | \overline{b}..\overline{q} | \overline{a − b + \frac{q+r−1}{2}}..\overline{r}]  ℓ + d + 1 ≥ q > ℓ + 1, r − q odd \frac{2k′−r−q+3}{2}
⊗[1..\frac{k′+b}{2} | \overline{\frac{k′+b}{2}}..\overline{b + 1}]  k′ − r odd, a − b < \frac{q−r−1}{2}

37

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Department of Mathematics, University of California, One Shields Avenue, Davis, CA 95616-8633, U.S.A.

E-mail address: anne@math.ucdavis.edu
URL: http://www.math.ucdavis.edu/~anne